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Limit inferior and limit superior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.

The limit inferior of a sequence $$(x_n)$$ is denoted by $$\liminf_{n\to\infty}x_n\quad\text{or}\quad \varliminf_{n\to\infty}x_n,$$ and the limit superior of a sequence $$(x_n)$$ is denoted by $$\limsup_{n\to\infty}x_n\quad\text{or}\quad \varlimsup_{n\to\infty}x_n.$$

Definition for sequences
The  of a sequence (xn) is defined by $$\liminf_{n\to\infty} x_n := \lim_{n\to\infty}\! \Big(\inf_{m \geq n} x_m\Big)$$ or $$\liminf_{n\to\infty} x_n := \sup_{n \geq 0}\,\inf_{m \geq n} x_m = \sup\,\{\, \inf\,\{\,x_m : m \geq n\,\} : n \geq 0\,\}.$$

Similarly, the  of (xn) is defined by $$\limsup_{n\to\infty} x_n := \lim_{n\to\infty}\! \Big(\sup_{m \geq n} x_m\Big)$$ or $$\limsup_{n\to\infty} x_n := \inf_{n \geq 0}\,\sup_{m \geq n} x_m = \inf\,\{\, \sup\,\{\,x_m : m \geq n\,\} : n \geq 0\,\}.$$

Alternatively, the notations $$\varliminf_{n\to\infty} x_n := \liminf_{n\to\infty} x_n$$ and $$\varlimsup_{n\to\infty} x_n := \limsup_{n\to\infty} x_n$$ are sometimes used.

The limits superior and inferior can equivalently be defined using the concept of subsequential limits of the sequence $$(x_n)$$. An element $$\xi$$ of the extended real numbers $$\overline{\R}$$ is a subsequential limit of $$(x_n)$$ if there exists a strictly increasing sequence of natural numbers $$(n_k)$$ such that $$\xi=\lim_{k\to\infty} x_{n_k}$$. If $$E \subseteq \overline{\R}$$ is the set of all subsequential limits of $$(x_n)$$, then
 * $$\limsup_{n\to\infty} x_n = \sup E $$

and
 * $$\liminf_{n\to\infty} x_n = \inf E.$$

If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i.e. the extended real number line) are complete. More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.

Whenever the ordinary limit exists, the limit inferior and limit superior are both equal to it; therefore, each can be considered a generalization of the ordinary limit which is primarily interesting in cases where the limit does not exist. Whenever lim inf xn and lim sup xn both exist, we have

 * $$\liminf_{n\to\infty}x_n \leq \limsup_{n\to\infty}x_n.$$

The limits inferior and superior are related to big-O notation in that they bound a sequence only "in the limit"; the sequence may exceed the bound. However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e−n may actually be less than all elements of the sequence. The only promise made is that some tail of the sequence can be bounded above by the limit superior plus an arbitrarily small positive constant, and bounded below by the limit inferior minus an arbitrarily small positive constant.

The limit superior and limit inferior of a sequence are a special case of those of a function (see below).

The case of sequences of real numbers
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set [−∞,∞], which is a complete lattice.

Interpretation
Consider a sequence $$(x_n)$$ consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite).
 * The limit superior of $$x_n$$ is the smallest real number $$b$$ such that, for any positive real number $$\varepsilon$$, there exists a natural number $$N$$ such that $$x_nN$$. In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than $$b+\varepsilon$$.
 * The limit inferior of $$x_n$$ is the largest real number $$b$$ such that, for any positive real number $$\varepsilon$$, there exists a natural number $$N$$ such that $$x_n>b-\varepsilon$$ for all $$n>N$$. In other words, any number below the limit inferior is an eventual lower bound for the sequence. Only a finite number of elements of the sequence are less than $$b-\varepsilon$$.

Properties
The relationship of limit inferior and limit superior for sequences of real numbers is as follows: $$\limsup_{n\to\infty} \left(-x_n\right) = -\liminf_{n\to\infty} x_n$$

As mentioned earlier, it is convenient to extend $$\R$$ to $$[-\infty, \infty].$$ Then, $$\left(x_n\right)$$ in $$[-\infty, \infty]$$ converges if and only if $$\liminf_{n\to\infty} x_n = \limsup_{n\to\infty} x_n$$ in which case $$\lim_{n\to\infty} x_n$$ is equal to their common value. (Note that when working just in $$\R,$$ convergence to $$-\infty$$ or $$\infty$$ would not be considered as convergence.) Since the limit inferior is at most the limit superior, the following conditions hold $$\begin{alignat}{4} \liminf_{n\to\infty} x_n &=  \infty &&\;\;\text{ implies }\;\; \lim_{n\to\infty} x_n = \infty, \\[0.3ex] \limsup_{n\to\infty} x_n &= - \infty &&\;\;\text{ implies }\;\; \lim_{n\to\infty} x_n = - \infty. \end{alignat}$$

If $$I = \liminf_{n\to\infty} x_n$$ and $$S = \limsup_{n\to\infty} x_n$$, then the interval $$[I, S]$$ need not contain any of the numbers $$x_n,$$ but every slight enlargement $$[I - \epsilon, S + \epsilon],$$ for arbitrarily small $$\epsilon > 0,$$ will contain $$x_n$$ for all but finitely many indices $$n.$$ In fact, the interval $$[I, S]$$ is the smallest closed interval with this property. We can formalize this property like this: there exist subsequences $$x_{k_n}$$ and $$x_{h_n}$$ of $$x_n$$ (where $$k_n$$ and $$h_n$$ are increasing) for which we have $$\liminf_{n\to\infty} x_n + \epsilon>x_{h_n} \;\;\;\;\;\;\;\;\; x_{k_n} > \limsup_{n\to\infty} x_n - \epsilon$$

On the other hand, there exists a $$n_0\in\mathbb{N}$$ so that for all $$n \geq n_0$$ $$\liminf_{n\to\infty} x_n - \epsilon < x_n < \limsup_{n\to\infty} x_n + \epsilon$$

To recapitulate:

 * If $$\Lambda$$ is greater than the limit superior, there are at most finitely many $$x_n$$ greater than $$\Lambda;$$ if it is less, there are infinitely many.
 * If $$\lambda$$ is less than the limit inferior, there are at most finitely many $$x_n$$ less than $$\lambda;$$ if it is greater, there are infinitely many.

Conversely, it can also be shown that:

 * If there are infinitely many $$x_n$$ greater than or equal to $$\Lambda$$, then $$\Lambda$$ is lesser than or equal to the limit supremum; if there are only finitely many $$x_n$$ greater than $$\Lambda$$, then $$\Lambda$$ is greater than or equal to the limit supremum.
 * If there are infinitely many $$x_n$$ lesser than or equal to $$\lambda$$, then $$\lambda$$ is greater than or equal to the limit inferior; if there are only finitely many $$x_n$$ lesser than $$\lambda$$, then $$\lambda$$ is lesser than or equal to the limit inferior.

In general,$$\inf_n x_n \leq \liminf_{n\to\infty} x_n \leq \limsup_{n\to\infty} x_n \leq \sup_n x_n.$$The liminf and limsup of a sequence are respectively the smallest and greatest cluster points.
 * For any two sequences of real numbers $$(a_n), (b_n),$$ the limit superior satisfies subadditivity whenever the right side of the inequality is defined (that is, not $$\infty - \infty$$ or $$-\infty + \infty$$): $$\limsup_{n\to\infty}\, (a_n + b_n) \leq \limsup_{n\to\infty} a_n +\ \limsup_{n\to\infty} b_n.$$

Analogously, the limit inferior satisfies superadditivity:$$\liminf_{n\to\infty}\, (a_n + b_n) \geq \liminf_{n\to\infty} a_n +\ \liminf_{n\to\infty} b_n.$$In the particular case that one of the sequences actually converges, say $$a_n \to a,$$ then the inequalities above become equalities (with $$\limsup_{n\to\infty} a_n$$ or $$\liminf_{n\to\infty} a_n$$ being replaced by $$a$$).

hold whenever the right-hand side is not of the form $$0 \cdot \infty.$$
 * For any two sequences of non-negative real numbers $$(a_n), (b_n),$$ the inequalities $$\limsup_{n\to\infty}\, (a_n b_n) \leq \left(\limsup_{n\to\infty} a_n \!\right) \!\!\left(\limsup_{n\to\infty} b_n \!\right)$$ and $$\liminf_{n\to\infty}\, (a_n b_n) \geq \left(\liminf_{n\to\infty} a_n \right)\!\!\left(\liminf_{n\to\infty} b_n\right)$$

If $$\lim_{n\to\infty} a_n = A$$ exists (including the case $$A = +\infty$$) and $$B = \limsup_{n\to\infty} b_n,$$ then $$\limsup_{n\to\infty} \left(a_n b_n\right) = A B$$ provided that $$A B$$ is not of the form $$0 \cdot \infty.$$

Examples

 * As an example, consider the sequence given by the sine function: $$x_n = \sin(n).$$ Using the fact that π is irrational, it follows that $$\liminf_{n\to\infty} x_n = -1$$ and $$\limsup_{n\to\infty} x_n = +1.$$ (This is because the sequence $$\{1, 2, 3, \ldots\}$$ is equidistributed mod 2π, a consequence of the equidistribution theorem.)

 * An example from number theory is $$\liminf_{n\to\infty}\, (p_{n+1} - p_n),$$ where $$p_n$$ is the $$n$$-th prime number.
 * The value of this limit inferior is conjectured to be 2 – this is the twin prime conjecture – but has only been proven to be less than or equal to 246. The corresponding limit superior is $$+\infty$$, because there are arbitrarily large gaps between consecutive primes.

Real-valued functions
Assume that a function is defined from a subset of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and −∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities). For example, given $$f(x) = \sin(1/x)$$, we have $$\limsup_{x\to 0} f(x) = 1$$ and $$\liminf_{x\to 0} f(x) = -1$$. The difference between the two is a rough measure of how "wildly" the function oscillates, and in observation of this fact, it is called the oscillation of f at 0. This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except on a set of measure zero. Note that points of nonzero oscillation (i.e., points at which f is "badly behaved") are discontinuities which, unless they make up a set of zero, are confined to a negligible set.

Functions from metric spaces
There is a notion of limsup and liminf for functions defined on a metric space whose relationship to limits of real-valued functions mirrors that of the relation between the limsup, liminf, and the limit of a real sequence. Take a metric space $$X$$, a subspace $$E$$ contained in $$X$$, and a function $$f:E \to \mathbb{R}$$. Define, for any limit point $$a$$ of $$E$$,

 * $$\limsup_{x \to a} f(x) = \lim_{\varepsilon \to 0} \left(\sup\,\{ f(x) : x \in E \cap B(a,\varepsilon)\setminus\{a\} \}\right)$$

and

 * $$\liminf_{x \to a} f(x) = \lim_{\varepsilon \to 0} \left(\inf\,\{ f(x) : x \in E \cap B(a,\varepsilon)\setminus\{a\} \}\right)$$

where $$B(a,\varepsilon)$$ denotes the metric ball of radius $$\varepsilon$$ about $$a$$.

Note that as ε shrinks, the supremum of the function over the ball is non increasing ( strictly decreasing or remaining the same), so we have

 * $$\limsup_{x \to a} f(x) = \inf_{\varepsilon > 0} \left(\sup\,\{ f(x) : x \in E \cap B(a,\varepsilon)\setminus\{a\} \}\right)$$

and similarly
 * $$\liminf_{x \to a} f(x) = \sup_{\varepsilon > 0} \left(\inf\,\{ f(x) : x \in E \cap B(a,\varepsilon)\setminus\{a\} \}\right).$$

Functions from topological spaces
This finally motivates the definitions for general topological spaces. Take X, E and a as before, but now let X be a topological space. In this case, we replace metric balls with neighborhoods:

 * $$\limsup_{x \to a} f(x) = \inf\,\{\, \sup\,\{ f(x) : x \in E \cap U\setminus\{a\} \} : U\ \mathrm{open},\, a \in U,\, E \cap U\setminus\{a\} \neq \emptyset \}$$
 * $$\liminf_{x \to a} f(x) = \sup\,\{\, \inf\,\{ f(x) : x \in E \cap U\setminus\{a\} \} : U\ \mathrm{open},\, a \in U,\, E \cap U\setminus\{a\} \neq \emptyset \}$$

(there is a way to write the formula using "lim" using nets and the neighborhood filter). This version is often useful in discussions of semi-continuity which crop up in analysis quite often. An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the closure of N in [−∞,∞], the extended real number line, is N ∪ {∞}.)

Sequences of sets
The power set ℘(X) of a set X is a complete lattice that is ordered by set inclusion, and so the supremum and infimum of any set of subsets (in terms of set inclusion) always exist. In particular, every subset Y of X is bounded above by X and below by the empty set ∅ because ∅ ⊆ Y ⊆ X. Hence, it is possible (and sometimes useful) to consider superior and inferior limits of sequences in ℘(X) (i.e., sequences of subsets of X).

There are two common ways to define the limit of sequences of sets. In both cases: The difference between the two definitions involves how the topology (i.e., how to quantify separation) is defined. In fact, the second definition is identical to the first when the discrete metric is used to induce the topology on X.
 * The sequence accumulates around sets of points rather than single points themselves. That is, because each element of the sequence is itself a set, there exist accumulation sets that are somehow nearby to infinitely many elements of the sequence.
 * The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points.
 * The infimum/inferior/inner limit is a set where all of these accumulation sets meet. That is, it is the intersection of all of the accumulation sets. When ordering by set inclusion, the infimum limit is the greatest lower bound on the set of accumulation points because it is contained in each of them. Hence, it is the infimum of the limit points.
 * Because ordering is by set inclusion, then the outer limit will always contain the inner limit (i.e., lim inf Xn ⊆ lim sup Xn). Hence, when considering the convergence of a sequence of sets, it generally suffices to consider the convergence of the outer limit of that sequence.

General set convergence
A sequence of sets in a metrizable space $$X$$ approaches a limiting set when the elements of each member of the sequence approach the elements of the limiting set. In particular, if $$(X_n)$$ is a sequence of subsets of $$X,$$ then: The limit $$\lim X_n$$ exists if and only if $$\liminf X_n$$ and $$\limsup X_n$$ agree, in which case $$\lim X_n = \limsup X_n = \liminf X_n.$$ The outer and inner limits should not be confused with the set-theoretic limits superior and inferior, as the latter sets are not sensitive to the topological structure of the space.
 * $$\limsup X_n,$$ which is also called the outer limit, consists of those elements which are limits of points in $$X_n$$ taken from (countably) infinitely many $$n.$$ That is, $$x \in \limsup X_n$$ if and only if there exists a sequence of points $$(x_k)$$ and a $$(X_{n_k})$$ of $$(X_n)$$ such that $$x_k \in X_{n_k}$$ and $$\lim_{k\to\infty} x_k = x.$$
 * $$\liminf X_n,$$ which is also called the inner limit, consists of those elements which are limits of points in $$X_n$$ for all but finitely many $$n$$ (that is, cofinitely many $$n$$). That is, $$x \in \liminf X_n$$ if and only if there exists a of points $$(x_k)$$ such that $$x_k \in X_k$$ and $$\lim_{k\to\infty} x_k = x.$$

Special case: discrete metric
This is the definition used in measure theory and probability. Further discussion and examples from the set-theoretic point of view, as opposed to the topological point of view discussed below, are at set-theoretic limit.

By this definition, a sequence of sets approaches a limiting set when the limiting set includes elements which are in all except finitely many sets of the sequence and does not include elements which are in all except finitely many complements of sets of the sequence. That is, this case specializes the general definition when the topology on set X is induced from the discrete metric.

Specifically, for points x, y ∈ X, the discrete metric is defined by
 * $$d(x,y) := \begin{cases} 0 &\text{if } x = y,\\ 1 &\text{if } x \neq y, \end{cases}$$

under which a sequence of points (xk) converges to point x ∈ X if and only if xk = x for all but finitely many k. Therefore, if the limit set exists it contains the points and only the points which are in all except finitely many of the sets of the sequence. Since convergence in the discrete metric is the strictest form of convergence (i.e., requires the most), this definition of a limit set is the strictest possible.

If (Xn) is a sequence of subsets of X, then the following always exist: Observe that x ∈ lim sup Xn if and only if x ∉ lim inf Xnc. In this sense, the sequence has a limit so long as every point in X either appears in all except finitely many Xn or appears in all except finitely many Xnc.
 * lim sup Xn consists of elements of X which belong to Xn for infinitely many n (see countably infinite). That is, x ∈ lim sup Xn if and only if there exists a subsequence (Xn k ) of (Xn) such that x ∈ Xn k for all k.
 * lim inf Xn consists of elements of X which belong to Xn for all except finitely many n (i.e., for cofinitely many n). That is, x ∈ lim inf Xn if and only if there exists some m > 0 such that x ∈ Xn for all n > m.
 * lim Xn exists if and only if lim inf Xn and lim sup Xn agree, in which case lim Xn = lim sup Xn = lim inf Xn.

Using the standard parlance of set theory, set inclusion provides a partial ordering on the collection of all subsets of X that allows set intersection to generate a greatest lower bound and set union to generate a least upper bound. Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf Xn, is the largest meeting of tails of the sequence, and the outer limit, lim sup Xn, is the smallest joining of tails of the sequence. The following makes this precise.
 * Let In be the meet of the nth tail of the sequence. That is,
 * $$\begin{align}I_n

&= \inf\,\{ X_m : m \in \{n, n+1, n+2, \ldots\}\}\\ &= \bigcap_{m=n}^{\infty} X_m = X_n \cap X_{n+1} \cap X_{n+2} \cap \cdots. \end{align}$$
 * The sequence (In) is non-decreasing (i.e. In ⊆ In+1) because each In+1 is the intersection of fewer sets than In. The least upper bound on this sequence of meets of tails is
 * $$\begin{align}

\liminf_{n\to\infty} X_n &= \sup\,\{ \,\inf\,\{X_m: m \in \{n, n+1, \ldots\}\}: n \in \{1,2,\dots\}\}\\ &= \bigcup_{n=1}^\infty \left({\bigcap_{m=n}^\infty}X_m\right)\!. \end{align}$$
 * So the limit infimum contains all subsets which are lower bounds for all but finitely many sets of the sequence.

 * Similarly, let Jn be the join of the nth tail of the sequence. That is,
 * $$\begin{align}J_n

&= \sup\,\{ X_m : m \in \{n, n+1, n+2, \ldots\}\}\\ &= \bigcup_{m=n}^{\infty} X_m = X_n \cup X_{n+1} \cup X_{n+2} \cup \cdots. \end{align}$$
 * The sequence (Jn) is non-increasing (i.e. Jn ⊇ Jn+1) because each Jn+1 is the union of fewer sets than Jn. The greatest lower bound on this sequence of joins of tails is
 * $$\begin{align}

\limsup_{n\to\infty} X_n &= \inf\,\{ \,\sup\,\{X_m: m \in \{n, n+1, \ldots\}\}: n \in \{1,2,\dots\}\}\\ &= \bigcap_{n=1}^\infty \left({\bigcup_{m=n}^\infty}X_m\right)\!. \end{align}$$
 * So the limit supremum is contained in all subsets which are upper bounds for all but finitely many sets of the sequence.

Examples
The following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set X.

 * Using the discrete metric

 * The Borel–Cantelli lemma is an example application of these constructs.

 * Using either the discrete metric or the Euclidean metric

 * Consider the set X = {0,1} and the sequence of subsets:
 * $$(X_n) = (\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots).$$
 * The "odd" and "even" elements of this sequence form two subsequences, ({0}, {0}, {0}, ...) and ({1}, {1}, {1}, ...), which have limit points 0 and 1, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the (Xn) sequence as a whole, and so the interior or inferior limit is the empty set { }. That is,
 * lim sup Xn = {0,1}
 * lim inf Xn = { }
 * However, for (Yn) = ({0}, {0}, {0}, ...) and (Zn) = ({1}, {1}, {1}, ...):
 * lim sup Yn = lim inf Yn = lim Yn = {0}
 * lim sup Zn = lim inf Zn = lim Zn = {1}

 * Consider the set X = {50, 20, −100, −25, 0, 1} and the sequence of subsets:
 * $$(X_n) = (\{50\}, \{20\}, \{-100\}, \{-25\}, \{0\}, \{1\}, \{0\}, \{1\}, \{0\}, \{1\}, \dots).$$
 * As in the previous two examples,
 * lim sup Xn = {0,1}
 * lim inf Xn = { }
 * That is, the four elements that do not match the pattern do not affect the lim inf and lim sup because there are only finitely many of them. In fact, these elements could be placed anywhere in the sequence. So long as the tails of the sequence are maintained, the outer and inner limits will be unchanged. The related concepts of essential inner and outer limits, which use the essential supremum and essential infimum, provide an important modification that "squashes" countably many (rather than just finitely many) interstitial additions.

 * Using the Euclidean metric

 * Consider the sequence of subsets of rational numbers:
 * $$(X_n) = ( \{0\}, \{1\}, \{1/2\}, \{1/2\}, \{2/3\}, \{1/3\}, \{3/4\}, \{1/4\}, \dots ).$$
 * The "odd" and "even" elements of this sequence form two subsequences, ({0}, {1/2}, {2/3}, {3/4}, ...) and ({1}, {1/2}, {1/3}, {1/4}, ...), which have limit points 1 and 0, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the (Xn) sequence as a whole, and so the interior or inferior limit is the empty set { }. So, as in the previous example,
 * lim sup Xn = {0,1}
 * lim inf Xn = { }
 * However, for (Yn) = ({0}, {1/2}, {2/3}, {3/4}, ...) and (Zn) = ({1}, {1/2}, {1/3}, {1/4}, ...):
 * lim sup Yn = lim inf Yn = lim Yn = {1}
 * lim sup Zn = lim inf Zn = lim Zn = {0}
 * In each of these four cases, the elements of the limiting sets are not elements of any of the sets from the original sequence.

 * The Ω limit (i.e., limit set) of a solution to a dynamic system is the outer limit of solution trajectories of the system. Because trajectories become closer and closer to this limit set, the tails of these trajectories converge to the limit set.
 * For example, an LTI system that is the cascade connection of several stable systems with an undamped second-order LTI system (i.e., zero damping ratio) will oscillate endlessly after being perturbed (e.g., an ideal bell after being struck). Hence, if the position and velocity of this system are plotted against each other, trajectories will approach a circle in the state space. This circle, which is the Ω limit set of the system, is the outer limit of solution trajectories of the system. The circle represents the locus of a trajectory corresponding to a pure sinusoidal tone output; that is, the system output approaches/approximates a pure tone.

Generalized definitions
The above definitions are inadequate for many technical applications. In fact, the definitions above are specializations of the following definitions.

Definition for a set
The limit inferior of a set X ⊆ Y is the infimum of all of the limit points of the set. That is,
 * $$\liminf X := \inf\,\{ x \in Y : x \text{ is a limit point of } X \}\,$$

Similarly, the limit superior of X is the supremum of all of the limit points of the set. That is,
 * $$\limsup X := \sup\,\{ x \in Y : x \text{ is a limit point of } X \}\,$$

Note that the set X needs to be defined as a subset of a partially ordered set Y that is also a topological space in order for these definitions to make sense. Moreover, it has to be a complete lattice so that the suprema and infima always exist. In that case every set has a limit superior and a limit inferior. Also note that the limit inferior and the limit superior of a set do not have to be elements of the set.

Definition for filter bases
Take a topological space X and a filter base B in that space. The set of all cluster points for that filter base is given by
 * $$\bigcap\, \{ \overline{B}_0 : B_0 \in B \}$$

where $$\overline{B}_0$$ is the closure of $$B_0$$. This is clearly a closed set and is similar to the set of limit points of a set. Assume that X is also a partially ordered set. The limit superior of the filter base B is defined as
 * $$\limsup B := \sup\, \bigcap\, \{ \overline{B}_0 : B_0 \in B \}$$

when that supremum exists. When X has a total order, is a complete lattice and has the order topology,
 * $$\limsup B = \inf\,\{ \sup B_0 : B_0 \in B \}.$$

Similarly, the limit inferior of the filter base B is defined as
 * $$\liminf B := \inf\, \bigcap\, \{ \overline{B}_0 : B_0 \in B \}$$

when that infimum exists; if X is totally ordered, is a complete lattice, and has the order topology, then
 * $$\liminf B = \sup\,\{ \inf B_0 : B_0 \in B \}.$$

If the limit inferior and limit superior agree, then there must be exactly one cluster point and the limit of the filter base is equal to this unique cluster point.

Specialization for sequences and nets
Note that filter bases are generalizations of nets, which are generalizations of sequences. Therefore, these definitions give the limit inferior and limit superior of any net (and thus any sequence) as well. For example, take topological space $$X$$ and the net $$(x_\alpha)_{\alpha \in A}$$, where $$(A,{\leq})$$ is a directed set and $$x_\alpha \in X$$ for all $$\alpha \in A$$. The filter base ("of tails") generated by this net is $$B$$ defined by
 * $$B := \{ \{ x_\alpha : \alpha_0 \leq \alpha \} : \alpha_0 \in A \}.\,$$

Therefore, the limit inferior and limit superior of the net are equal to the limit superior and limit inferior of $$B$$ respectively. Similarly, for topological space $$X$$, take the sequence $$(x_n)$$ where $$x_n \in X$$ for any $$n \in \mathbb{N}$$. The filter base ("of tails") generated by this sequence is $$C$$ defined by
 * $$C := \{ \{ x_n : n_0 \leq n \} : n_0 \in \mathbb{N} \}.\,$$

Therefore, the limit inferior and limit superior of the sequence are equal to the limit superior and limit inferior of $$C$$ respectively.

title: Limit inferior and limit superior


text:
Manuel II Palaiologos

Manuel II Palaiologos or Palaeologus (27 June 1350 – 21 July 1425) was Byzantine emperor from 1391 to 1425. Shortly before his death he was tonsured a monk and received the name Matthew. His wife Helena Dragaš saw to it that their sons, John VIII and Constantine XI, became emperors. He is commemorated by the Greek Orthodox Church on July 21.

Life

Manuel II Palaiologos was the second son of Emperor John V Palaiologos and his wife Helena Kantakouzene. Granted the title of despotēs by his father, the future Manuel II traveled west to seek support for the Byzantine Empire in 1365 and in 1370, serving as governor in Thessalonica from 1369. The failed attempt at usurpation by his older brother Andronikos IV Palaiologos in 1373 led to Manuel's being proclaimed heir and co-emperor of his father. He was crowned on 25 September 1373.

In 1376–1379 and again in 1390, Manuel and his father were supplanted by Andronikos IV and then his son John VII, but Manuel personally defeated his nephew with help from the Republic of Venice in 1390. Although John V had been restored, Manuel was forced to go as an honorary hostage to the court of the Ottoman Sultan Bayezid I at Prousa (Bursa). During his stay, Manuel was forced to participate in the Ottoman campaign that reduced Philadelpheia, the last Byzantine enclave in Anatolia.

Siege of Constantinople and letters to European courts
Having heard of his father's death in February 1391, Manuel II Palaiologos fled the Ottoman court and secured the capital against any potential claim by his nephew John VII. Following Manuel's coronation the Ottoman Sultan was initially content to leave Byzantium in comparative peace. However, in 1393 a large insurrection erupted in Bulgaria which, although successfully put down by the Ottomans, caused Bayezid to lapse into an episode of paranoia in which he believed his various Christian vassals were plotting against him. Bayezid called all his Christian vassals to a meeting at Serres, with the intention of massacring them, a decision he relented on only at the last moment. The episode is said to have left all of the Christian vassal rulers shaken and convinced Manuel that continued appeasement towards the Ottomans was not a guarantee of his own personal safety or the continued survival of the empire and that efforts must be made to obtain Western aid.

Sultan Bayezid I blockaded Constantinople from 1394 to 1402. In the meantime, an anti-Ottoman crusade led by the Hungarian King Sigismund of Luxemburg failed at the Battle of Nicopolis on 25 September 1396. Manuel II had sent 10 ships to help in that Crusade. In October 1397, Theodore Kantakouzenos, Manuel's uncle, alongside John of Natala arrived at the court of Charles VI of France, bearing the Emperor's letters (dated 1 July 1397) requesting the French king's military aid. In addition, Charles also provided funds for the two nobles to treat with King Richard II of England in April 1398, with the aim of soliciting further aid. Though the latter was preoccupied by domestic troubles at this point to provide any support.

However, the two nobles returned home with the Marshal of France Jean II Le Maingre who was sent from Aigues-Mortes with six ships carrying 1,200 men to assist Manuel II. The Marshal encouraged the latter to go personally to seek assistance against the Ottoman Empire from the courts of western Europe. After some five years of siege, Manuel II entrusted the city to his nephew, aided by a French garrison of 300 men led by Seigneur Jean de Châteaumorand and embarked (along with a suite of 40 people) on a long trip abroad along with the Marshal.

Emperor's trip to the West
On 10 December 1399, Manuel II embarked on his trip by starting to sail to the Morea, where he left his wife and children with his brother Theodore I Palaiologos to be protected from his nephew's intentions. He later landed in Venice in April 1400, then he went to Padua, Vicenza and Pavia, until he reached Milan, where he met Duke Gian Galeazzo Visconti, and his close friend Manuel Chrysoloras. Afterwards, he met Charles VI of France at Charenton on 3 June 1400. During his stay in France, Manuel II continued to contact European monarchs.

According to Michel Pintoin who chronicled the visit to Paris: "Then, the king raised his hat, and the emperor raised his imperial cap – he had no hat – and both greeted one another in the most honourable way. When he had welcomed [the emperor], the king accompanied him into Paris, riding side by side. They were followed by the Princes of the Blood who, once the banquet in the royal palace finished, escorted [the emperor] to the lodgings which had been prepared for him in the Louvre castle]."

In December 1400, he embarked to England to meet Henry IV of England who received him at Blackheath on the 21st of that month, making him the only Byzantine emperor ever to visit England, where he stayed at Eltham Palace until mid-February 1401, and a joust took place in his honour. In addition, he received £2,000, in which he acknowledged receipt of the funds in a Latin document and sealed it with his own golden bull.

Thomas Walsingham wrote about Manuel II's visit to England: "At the same time the Emperor of Constantinople visited England to ask for help against the Turks. The king with an imposing retinue, met him at Blackheath on the feast of St Thomas [21 December], gave so great a hero an appropriate welcome and escorted him to London. He entertained him there royally for many days, paying the expenses of the emperor's stay, and by grand presents showing respect for a person of such eminence."

Moreover, Adam of Usk reported: "On the feast of St Thomas the apostle [21 December], the emperor of the Greeks visited the king of England in London to seek help against the Saracens, and was honourably received by him, staying with him for two whole months at enormous expense to the king, and being showered with gifts at his departure. This emperor and his men always went about dressed uniformly in long robes cut like tabards which were all of one colour, namely white, and disapproved greatly of the fashions and varieties of dress worn by the English, declaring that they signified inconstancy and fickleness of heart. No razor ever touched the heads or beards of his priests. These Greeks were extremely devout in their religious services, having them chanted variously by knights or by clerics, for they were sung in their native tongue. I thought to myself how sad it was that this great Christian leader from the remote east had been driven by the power of the infidels to visit distant islands in the west in order to seek help against them."

However, Manuel II sent a letter to his friend Manuel Chrysoloras, describing his visit to England: "Now what is the reason for the present letter? A large number of letters have come to us from all over bearing excellent and wonderful promises, but most important is the ruler with whom we are now staying, the king of Britain the Great, of a second civilized world, you might say, who abounds in so many good qualities and is adorned with all sorts of virtues. His reputation earns him the admiration of people who have not met him, while for those who have once seen him, he proves brilliantly that Fame is not really a goddess, since she is unable to show the man to be as great as does actual experience. This ruler, then, is most illustrious because of his position, most illustrious too, because of his intelligence; his might amazes everyone, and his understanding wins him friends; he extends his hand to all and in every way he places himself at the service of those who need help. And now, in accord with his nature, he has made himself a virtual haven for us in the midst of a twofold tempest, that of the season and that of fortune, and we have found refuge in the man himself and his character. His conversation is quite charming; he pleases us in every way; he honours us to the greatest extent and loves us no less. Although he has gone to extremes in all he has done for us, he seems almost to blush in the belief—in this he is alone—that he might have fallen considerably short of what he should have done. This is how magnanimous the man is."

Manuel II later returned to France with high hopes of substantial help and funds for Constantinople. In the meantime, he sent delegations with relics including pieces of the tunic of Christ and a piece of the Holy Sponge to Pope Boniface IX and Antipope Benedict XIII, Queen Margaret I of Denmark, king Martin of Aragon and king Charles III of Navarre to seek further assistance. He eventually left France on 23 November 1402, and finally returned to Constantinople in June 1403.

Renewed Ottoman sieges
The Ottomans under Bayezid I were themselves crushingly defeated by Timur at the Battle of Ankara in 1402. As the sons of Bayezid I struggled with each other over the succession in the Ottoman Interregnum, John VII was able to secure the return of the European coast of the Sea of Marmara and of Thessalonica to the Byzantine Empire in the Treaty of Gallipoli. When Manuel II returned home in 1403, his nephew duly surrendered control of Constantinople and received as a reward the governorship of newly recovered Thessalonica. The treaty also regained from the Ottomans Mesembria (1403–1453), Varna (1403–1415), and the Marmara coast from Scutari to Nicomedia (between 1403–1421).

However, Manuel II kept contact with Venice, Genoa, Paris and Aragon, by sending envoy Manuel Chrysoloras in 1407–8, pursuing to form a coalition against the Ottomans.

On 25 July 1414, with a fleet consisting of four galleys and two other vessels carrying contingents of infantry and cavalry, departed Constantinople for Thessalonica. The purpose of this force soon became clear when he made an unannounced stop at Thasos, a normally unimportant island which was then under threat from a son of the lord of Lesbos, Francesco Gattilusio. It took Manuel three months to reassert imperial authority on the island. Only then did he continue on to Thessalonica, where he was warmly met by his son Andronicus, who then governed the city.

In the spring of 1415, he and his soldiers left for the Peloponnese, arriving at the little port of Kenchreai on Good Friday, 29 March. Manuel II Palaiologos used his time there to bolster the defences of the Despotate of Morea, where the Byzantine Empire was actually expanding at the expense of the remnants of the Latin Empire. Here Manuel supervised the building of the Hexamilion (six-mile wall) across the Isthmus of Corinth, intended to defend the Peloponnese from the Ottomans.

Manuel II stood on friendly terms with the victor in the Ottoman civil war, Mehmed I (1402–1421), but his attempts to meddle in the next contested succession led to a new assault on Constantinople by Murad II (1421–1451) in 1422. During the last years of his life, Manuel II relinquished most official duties to his son and heir John VIII Palaiologos, and went back to the West searching for assistance against the Ottomans, this time to the King Sigismund of Hungary, staying for two months in his court of Buda. Sigismund (after suffering a defeat against the Turks in the Battle of Nicopolis in 1396) never rejected the possibility of fighting against the Ottoman Empire. However, with the Hussite wars in Bohemia, it was impossible to count on the Czech or German armies, and the Hungarian ones were needed to protect the Kingdom and control the religious conflicts. Unhappily Manuel returned home with empty hands from the Hungarian Kingdom, and in 1424 he and his son were forced to sign an unfavourable peace treaty with the Ottoman Turks, whereby the Byzantine Empire had to pay tribute to the sultan.

Death
Manuel II was paralyzed by a stroke on 1 October 1422, but his mind was unaffected and he continued to rule for three more years. He lived his last few days as a monk, taking the name of Matthew. He died on 21 July 1425, aged 75, and was buried at the Pantokrator Monastery in Constantinople.

Writings
Manuel II was the author of numerous works of varied character, including letters, poems, a Saint's Life, treatises on theology and rhetoric, and an epitaph for his brother Theodore I Palaiologos and a mirror of princes for his son and heir John. This mirror of princes has special value, because it is the last sample of this literary genre bequeathed to us by Byzantines.

Family
By his wife Helena Dragas, the daughter of the Serbian prince Constantine Dragas, Manuel II Palaiologos had several children, including:
 * A daughter. Mentioned as the eldest daughter but not named.
 * Constantine Palaiologos. Born c. 1393/8, died before 1405 in Monemvasia.
 * John VIII Palaiologos (18 December 1392 – 31 October 1448). Byzantine emperor, 1425–1448.
 * Andronikos Palaiologos, Lord of Thessalonica (d. 1429).
 * A second daughter. Also not named in the text.
 * Theodore II Palaiologos, Lord of Morea (d. 1448).
 * Michael Palaiologos. Born 1406/7, died 1409/10 of the plague.
 * Constantine XI Dragases Palaiologos (8 February 1405 – 29 May 1453). Despotēs in the Morea and subsequently the last Byzantine emperor, 1448–1453.
 * Demetrios Palaiologos (c. 1407–1470). Despotēs in the Morea.
 * Thomas Palaiologos (c. 1409 – 12 May 1465). Despotēs in the Morea.

Pope Benedict XVI controversy
In a lecture delivered on 12 September 2006, Pope Benedict XVI quoted from a dialogue believed to have occurred in 1391 between Manuel II and a Persian scholar and recorded in a book by Manuel II (Dialogue 7 of Twenty-six Dialogues with a Persian) in which the Emperor stated: "Show me just what Muhammad brought that was new and there you will find things only evil and inhuman, such as his command to spread by the sword the faith he preached."

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Manuel II Palaiologos


text:
Deixis

In linguistics, deixis is the use of words or phrases to refer to a particular time (e.g. then), place (e.g. here), or person (e.g. you) relative to the context of the utterance. Deixis exists in all known natural languages and is closely related to anaphora, with a sometimes unclear distinction between the two. In linguistic anthropology, deixis is seen as the same as, or a subclass of, indexicality.

The term's origin is δεῖξις. To this, Chrysippus (c. 279) added the specialized meaning point of reference, which is the sense in which the term is used in contemporary linguistics.

Types
There are three main types of deictic words, as described by Charles J. Fillmore: personal, spatial, and temporal. In some languages, these may overlap, such as spatial and personal deixis in many signed pronouns. Some linguists consider social deixis to be a fourth type.

Personal
Personal deictic words, called personal pronouns in English, refer to the grammatical persons involved in an utterance. These can include the first person (speaker), second person (addressee), third, and in some languages fourth and fifth person. Personal deixis may give further information about the referent, such as gender. Examples examples of personal deixis include:

 * I am going to the cinema.
 * Would you like to have dinner?
 * They tried to hurt me, but she helped me.

Spatial
Spatial, or place, deixis is used to refer to spatial locations relative to an utterance. Similarly to personal deixis, the locations may be either those of the speaker and addressee or those of persons or objects being referred to. Spatial demonstratives include locative adverbs (e.g. here and there) and demonstratives (e.g. this, these, that, and those) although those are far from exclusive. Spatial demonstratives are often relative to the location of the speaker such as:

 * The shop is across the street.

where "across the street" is understood to mean "across the street from where I [the speaker] am right now."

Words relating to spatial deixis can be proximal (near, such as English [right] here or this), medial (near the addressee, such as English [over] there or that), distal (far, such as English [out] there or that), far-distal (far from both the speaker and addressee, such as archaic English yon and yonder). The Malagasy language has seven degrees of distance combined with two degrees of visibility, while many Inuit languages have even more complex systems.

Temporal
Temporal, or time, deixis is used to refer to time relevant to the utterance. This includes temporal adverbs (e.g. then and soon), nouns (e.g. tomorrow) and use of grammatical tense. Temporal deixis can can be relative to the time when an utterance is made (the speaker’s "now") or the time when the utterance is heard or seen (the addressee’s "now"). Although these are often the same time, they can differ in cases such as a voice recording or written text. For example:

 * It is raining now, but I hope when you read this it will be sunny.

Tenses are usually separated into absolute (deictic) and relative tenses. For example, simple English past tense is absolute, such as "He went." whereas the pluperfect is relative to some other deictically specified time, as in "When I got home, he had gone."

Discourse deixis
Discourse deixis, also referred to as text deixis, refers to the use of expressions within an utterance to refer to parts of the discourse that contain the utterance—including the utterance itself. For example, in "This is a great story." this refers to an upcoming portion of the discourse.

Switch reference is a type of discourse deixis, and a grammatical feature found in some languages, which indicates whether the argument of one clause is the same as the argument of the previous clause. In some languages, this is done through same subject markers and different subject markers. In the translated example "John punched Tom, and left-[same subject marker]," it is John who left, and in "John punched Tom, and left-[different subject marker]," it is Tom who left.

Discourse deixis has been observed in internet language, particularly with the use of iconic language forms resembling arrows.

Social deixis
Social deixis concerns the social information that is encoded within various expressions, such as relative social status and familiarity. These include T–V distinctions and honorifics.

Deictic center
A deictic center, sometimes referred to as an origo, is a set of theoretical points that a deictic expression is 'anchored' to, such that the evaluation of the meaning of the expression leads one to the relevant point. As deictic expressions are frequently egocentric, the center often consists of the speaker at the time and place of the utterance and, additionally, the place in the discourse and relevant social factors. However, deictic expressions can also be used in such a way that the deictic center is transferred to other participants in the exchange or to persons / places / etc. being described in a narrative. So, for example, in the sentence;

 * I am standing here now.

the deictic center is simply the person at the time and place of speaking. But say two people are talking on the phone long-distance, from London to New York. The Londoner can say,

 * We are leaving [London] next week.

in which case the deictic center is in London, or they can equally validly say,

 * We are coming [to New York] next week.

in which case the deictic center is in New York. Similarly, when telling a story about someone, the deictic center is likely to switch to him, her or they (third-person pronouns). So then in the sentence;

 * He then ran twenty feet to the left.

it is understood that the center is with the person being spoken of, and thus, "to the left" refers not to the speaker's left, but to the object of the story's left, that is, the person referred to as 'he' at the time immediately before he ran twenty feet.

Usages
It is helpful to distinguish between two usages of deixis, gestural and symbolic, as well as non-deictic usages of frequently deictic words. Gestural deixis refers, broadly, to deictic expressions whose understanding requires some sort of audio-visual information. A simple example is when an object is pointed at and referred to as "this" or "that". However, the category can include other types of information than pointing, such as direction of gaze, tone of voice, and so on. Symbolic usage, by contrast, requires generally only basic spatio-temporal knowledge of the utterance. So, for example

 * I broke this finger.

requires being able to see which finger is being held up, whereas

 * I love this city.

requires only knowledge of the current location. In a similar vein,

 * I went to this city one time ...

is a non-deictic usage of "this", which does not identify anywhere specifically. Rather, it is used as an indefinite article, much the way "a" could be used in its place.

Distinction with similar terms
The distinction between deixis and anaphora is unclearly defined. Generally, an anaphoric reference refers to something within a text that has been previously identified. For example, in "Susan dropped the plate. It shattered loudly," the word it refers to the phrase, "the plate". An expression can be both deictic and anaphoric at the same time, for example "I was born in London, and I have lived here/there all my life." here or there function anaphorically in their reference to London, and deictically in that the choice between "here" or "there" indicates whether the speaker is or is not currently in London.

The terms deixis and indexicality are frequently used almost interchangeably, and both deal with essentially the same idea of contextually-dependent references. However, the two terms have different histories and traditions. In the past, deixis was associated specifically with spatiotemporal reference, and indexicality was used more broadly. More importantly, each is associated with a different field of study. Deixis is associated with linguistics, and indexicality is associated with philosophy as well as pragmatics.

title: Deixis


text:
Alessandro Manzoni

Alessandro Francesco Tommaso Antonio Manzoni (,, ; 7 March 1785 – 22 May 1873) was an Italian poet, novelist and philosopher. He is famous for the novel The Betrothed (orig. I promessi sposi) (1827), generally ranked among the masterpieces of world literature. The novel is also a symbol of the Italian Risorgimento, both for its patriotic message and because it was a fundamental milestone in the development of the modern, unified Italian language. Manzoni also contributed to the stabilization of the modern Italian language and helped to ensure linguistic unity throughout Italy. He was an influential proponent of Liberal Catholicism in Italy. His work and thinking has often been contrasted with that of his younger contemporary Giacomo Leopardi by critics.

Early life
Manzoni was born in Milan, Italy, on 7 March 1785. Pietro, his father, aged about fifty, belonged to an old family of Lecco, originally feudal lords of Barzio, in the Valsassina. However, his biological father was likely the illuminist Alessandro Verri, brother of the influential writer Pietro Verri. The poet's maternal grandfather, Cesare Beccaria, was a well-known author and philosopher, and his mother Giulia had literary talent as well. The young Alessandro spent his first two years in cascina Costa in Galbiate and he was wet-nursed by Caterina Panzeri, as attested by a memorial tablet affixed in the place. In 1792 his parents broke their marriage and his mother began a relationship with the writer Carlo Imbonati, moving to England and later to Paris. For this reason, Alessandro was brought up in several religious institutions.

Manzoni was a slow developer, and at the various colleges he attended he was considered a dunce. At fifteen, however, he developed a passion for poetry and wrote two sonnets of considerable merit. Upon the death of his father in 1807, he joined the freethinking household of his mother at Auteuil, and spent two years mixing with the literary set of the so-called "ideologues", philosophers of the 18th-century school, among whom he made many friends, notably Claude Charles Fauriel. At Auteuil, he developed a lifelong interest in liberalism. He was even supposed to marry the daughter of Antoine Destutt de Tracy. There too he imbibed the anti-Catholic creed of Voltairianism.

In 1806–1807, while at Auteuil, he first appeared before the public as a poet, with two works, one entitled Urania, in the classical style, of which he became later the most conspicuous adversary, the other an elegy in blank verse, on the death of Count Carlo Imbonati, from whom, through his mother, he inherited considerable property, including the villa of Brusuglio, thenceforth his principal residence.

1808–1821
In 1808, Manzoni married Henriette Blondel, daughter of a Genevese banker. She came from a Calvinist family, but in 1810 she became a Roman Catholic. Her conversion profoundly influenced her husband. That same year he experienced a religious crisis which led him from Jansenism to an austere form of Catholicism. Manzoni's marriage proved a happy one, and he led for many years a retired domestic life, divided between literature and the picturesque husbandry of Lombardy.

His intellectual energy in this period of his life was devoted to the composition of the Inni sacri, a series of sacred lyrics, and of a treatise on Catholic morality, Osservazioni sulla morale cattolica, a task undertaken under religious guidance, in reparation for his early lapse from faith. In 1818 he had to sell his paternal inheritance, as his money had been lost to a dishonest agent. His characteristic generosity was shown at this time in his dealings with his peasants, who were heavily indebted to him. He not only cancelled on the spot the record of all sums owed to him, but bade them keep for themselves the whole of the coming maize harvest.

In 1819, Manzoni published his first tragedy, Il Conte di Carmagnola, which, boldly violating all classical conventions, excited a lively controversy. It was severely criticized in a Quarterly Review article to which Goethe replied in its defence, "one genius," as Count de Gubernatis remarks, "having divined the other." The death of Napoleon in 1821 inspired Manzoni's powerful stanzas Il Cinque maggio (The Fifth of May), one of the most popular lyrics in the Italian language. The political events of that year, and the imprisonment of many of his friends, weighed much on Manzoni's mind, and the historical studies in which he sought distraction during his subsequent retirement at Brusuglio suggested his great work.

The Betrothed
Manzoni started work on the novel in 1821, but he began the actual composition of Fermo e Lucia on 24 April 1821, after reading the novels of Walter Scott, mainly in French translations. Round the episode of the Innominato, historically identified with Bernardino Visconti, the first manuscript of the novel The Betrothed (in Italian I promessi sposi) began to grow into shape, and was completed in September 1823. The work was published, after being deeply reshaped by the author and revised by friends in 1825–1827, at the rate of a volume a year; it at once raised its author to the first rank of literary fame. It is generally agreed to be his greatest work, and the paradigm of modern Italian language. The Penguin Companion to European Literature notes that 'the book's real greatness lies in its delineation of character...in the heroine, Lucia, in Padre Cristoforo, the Capuchin friar, and the saintly cardinal (Borromeo) of Milan, he has created three living examples of that pure and wholehearted Christianity which is his ideal. But his psychological penetration extends also to those who fall short of this standard, whether through weakness or perversity, and the novel is rich in pictures of ordinary men and women, seen with a delightful irony and disenchantment which always stops short of cynicism, and which provides a perfect balance for the evangelical fervour of his ideal'. In 1822, Manzoni published his second tragedy, Adelchi, turning on the overthrow by Charlemagne of the Lombard domination in Italy, and containing many veiled allusions to the existing Austrian rule. With these works Manzoni's literary career was practically closed. But he laboriously revised The Betrothed in Tuscan-Italian, and in 1840 republished it in that form, with a historical essay, Storia della colonna infame, on details of the 17th-century plague in Milan so important in the novel. He also wrote a small treatise on the Italian language.

Politics and economics
Manzoni favored the Italian unification and was even a member of the Italian Senate. Before and after his embracing an austere Catholicism upon marrying Henriett Blondel, Manzoni's politics can be broadly described as liberal. Since his French trip, Manzoni's liberalism included a profound understanding of economics. He was well acquainted with authors such as Jean-Baptiste Say and Adam Smith and left numerous notes on the economic treatises and essays he was reading.

His understanding of economics came to surface in his grand historical novel The Betrothed, particularly in Chapter 12, where he deals with the famine in Lombardy. Economist and President of the Italian Republic Luigi Einaudi praised the chapter and the whole of The Betrothed as "one of the best treatises on political economy ever written". Economic historian Deirdre N. McCloskey likewise described it as "a lecture in Economics 101".

Family, death and legacy

The death of Manzoni's wife in 1833 was preceded and followed by those of several of his children, and of his mother. In the mid-1830s he attended the "Salotto Maffei", a salon in Milan hosted by Clara Maffei, and in 1837 he married again, to Teresa Borri, widow of Count Stampa. Teresa also died before him, while of nine children born to him in his two marriages all but two pre-deceased him. In 1860 King Victor Emmanuel II named him a senator. The death of his eldest son, Pier Luigi, on 28 April 1873, was the final blow which hastened his end. He was already weakened as he had fallen on 6 January while exiting the San Fedele church, hitting his head on the steps, and he died after 5 months of cerebral meningitis, a complication of the trauma. His funeral was celebrated in the Milan Cathedral with almost royal pomp. His remains, after they lay in state for some days, were followed to the Cimitero Monumentale in Milan by a vast cortege, including the royal princes and all the great officers of state, but his noblest monument was Giuseppe Verdi's Requiem, written to honour his memory. In modern times, he has been honored twice with a Google Doodle.

His Osservazioni sulla morale cattolica was quoted by Pope Pius XI in his encyclical on Christian Education Divini Illius Magistri: "'20. It is worthy of note how a layman, an excellent writer and at the same time a profound and conscientious thinker, has been able to understand well and express exactly this fundamental Catholic doctrine: 'The Church does not say that morality belongs purely, in the sense of exclusively, to her; but that it belongs wholly to her. She has never maintained that outside her fold and apart from her teaching, man cannot arrive at any moral truth; she has on the contrary more than once condemned this opinion because it has appeared under more forms than one. She does however say, has said, and will ever say, that because of her institution by Jesus Christ, because of the Holy Ghost sent her in His name by the Father, she alone possesses what she has had immediately from God and can never lose, the whole of moral truth, omnem veritatem, in which all individual moral truths are included, as well those which man may learn by the help of reason, as those which form part of revelation or which may be deduced from it''."

title: Alessandro Manzoni


text:
1730s BC

The 1730s BC was a decade lasting from January 1, 1739 BC to December 31, 1730 BC.

Events and trends

 * 1736 BC–According to the ultra-long chronology of the ancient Near East, this is the year the sack of Babylon occurred.

Significant people

 * Samsu-iluna, king of Babylon since 1750 BC, according to the middle chronology
 * Rim-Sin I, ruler of the Middle Eastern city-state of Larsa since 1758 BC, according to the short chronology. That same chronology records his death in 1699 BC.

title: 1730s BC


text:
User:JaWi

Hey, I'm listed in Wikipedia!

Well, since we're on the topic: who am I?

I'm a 24yo ex-student from the Netherlands majored in Telematics and currently looking for a permanent job (or something similar ;-))

That's it for the moment...

C'ya

-- JaWi

title: User:JaWi