Boolean differential calculus

Upper and lower limit of a real sequence

Definition

The upper and lower limit of a sequence of real numbers $\{x_n\}$ (called also limes superior and limes inferior) can be defined in several ways and are denoted, respectively as \[ \limsup_{n\to\infty}\, x_n\qquad \liminf_{n\to\infty}\,\, x_n \] (some authors use also the notation $\overline{\lim}$ and $\underline{\lim}$). One possible definition is the following

Definition 1 \[ \limsup_{n\to\infty} \, x_n = \inf_{n\in\mathbb N}\,\, \sup_{k\geq n}\, x_k \] \[ \liminf_{n\to\infty}\,<, x_n = \sup_{n\in\mathbb N}\,\, \inf_{k\geq n}\, x_k\, . \]

Properties

It follows easily from the definition that \[ \liminf_n\,\, x_n = -\limsup_n\, (-x_n)\, , \] \[ \liminf_n\,\, (\lambda x_n) = \lambda\, \liminf_n\,\, x_n\qquad \limsup_n\, (\lambda x_n) = \lambda\, \limsup_n\, x_n\qquad \mbox{when '"`UNIQ-MathJax4-QINU`"'} \] and that \[ \liminf_n\,\, (x_n + y_n)\geq \liminf\, x_n + \liminf\,\, y_n \qquad \limsup_n\, (x_n + y_n)\leq \limsup\, x_n + \limsup\, y_n \] if the additions are not of the type $-\infty + \infty$.

Moreover, the limit of $\{x_n\}$ exists and it is a real number $L$ (respetively $\infty$, $-infty$) if and only if the upper and lower limit coincide and are a real number $L$ (resp. $\infty$, $-\infty$).

The upper and lower limits of a sequence are both finite if and only if the sequence is bounded.

Characterizations

The upper and lower limits can also be defined in several alternative ways. In particular

Theorem 1 Let $S:=\{a\in ]-\infty, \infty] : \{k: x_k >a\} \mbox{is finite}\}$ and $L:= \{a\in [-\infty, \infty[ : \{k: x_k <a\} \mbox{is finite}\}$. Then $\limsup x_n$ is the minimum of $S$ and $\liminf x_n$ is the maximum of $L$.

Theorem 2 Consider the set $A$ of elements $\ell\in [-\infty, \infty]$ for which there is a subsequence of $\{y_n\}$ converging to $\ell$. Then $\limsup x_n$ is the maximum of $A$ and $\liminf x_n$ is the minimum of $A$.

Theorem 3 $U:=\limsup x_n$ is characterized by the two properties:

• if $U< \infty$ for all $u> U$ there is $N\in \mathbb N$ such that $x_n< u$ for all $n> N$;
• if $U> -\infty$ for all $u< U$ and $N\in \mathbb N$ there is a $k>N$ with $x_k> u$.

$L:=\liminf x_n$ is characterized by the two properties:

• if $U> -\infty$ for all $u< U$ there is $N\in\mathbb N$ such that $x_n> u$ for all $n> N$;
• if $U< \infty$ for all $u> U$ and $N\in\mathbb N$ there is a $k> N$ with $x_k< u$.

Examples

If $x_n = (-1)^n$ then \[ \liminf\, x_n = -1 \qquad \mbox{and} \qquad \limsup_n\, x_n = 1\, . \] If $x_n = (-1)^n n$ then \[ \liminf\, x_n = -\infty \qquad \mbox{and} \qquad \limsup_n\, x_n = \infty\, . \] If $x_n = n + (-1)^n$, then \[ \liminf\, x_n = 0 \qquad \mbox{and} \qquad \limsup_n\, x_n = \infty\, . \]

Upper and lower limit of a real function

Definition

If $f$ is a real-valued function defined on a set $E\subset \mathbb R$ (or $\subset \mathbb R^k$), the upper and lower limits of $f$ at $x_0$ are denoted by \[ \limsup_{x\to x_0}\, f(x)\qquad \mbox{and}\qquad \liminf_{x\to x_0}\, f(x)\, . \] and, under the assumptions that $x_0$ is an accumulation point for $E$ (i.e. there is a sequence $\{y_n\}\subset E\setminus x_0\}$ converging to $x_0$) can be defined as

Definition 4 \[ \limsup_{x\to x_0}\, f(x) = \inf_{r> 0} \,\sup\, \{f(x): |x-x_0|< r, x\in E \setminus \{x_0\}\} \] \[ \liminf_{x\to x_0}\, f(x) = \sup_{r> 0} \,\inf\, \{f(x): |x-x_0|< r, x\in E \setminus \{x_0\}\} \]

(Some authors include also the point $x_0$ in the definitions above, however this choice is less common).

The definition above can be easilily extended to functions defined on an arbitrary metric space $(X, d)$: it suffices to replace $|x-x_0|< r$ with $d (x, x_0)< r$, namely

Definition 5 \[ \limsup_{x\to x_0}\, f(x) = \inf_{r> 0} \,\sup\, \{f(x): d(x,x_0)< r, x\in E \setminus \{x_0\}\} \] \[ \liminf_{x\to x_0}\, f(x) = \sup_{r> 0} \,\inf\, \{f(x): d(x,x_0)< r, x\in E \setminus \{x_0\}\} \]

As in the case of sequences, some authors use the notation $\overline{\lim}$ and $\underline{\lim}$.

Characterizations

The upper and lower limits \[ U =\limsup_{x\to x_0}\, f(x) \] \[ L :=\liminf_{x\to x_0}\, f(x) \] can also be defined in several alternative ways. A useful one, which reduces to sequences, is the following:

Theorem 6 $U$ is characterized by the properties:

• There is a sequence $\{y_k\}\subset E\setminus \{x_0\}$ such that $\lim y_k = x_0$ and $\lim f(y_k) = U$;
• For any sequence $\{y_k\}\subset E\setminus \{x_0\}$ converging to $x_0$ we have $\limsup\, f (y_k)\leq U$.

$L$ is characterized by the properties:

• There is a sequence $\{y_k\}\subset E\setminus \{x_0\}$ such that $\lim y_k = x_0$ and $\lim f(y_k) = L$;
• For any sequence $\{y_k\}\subset E\setminus \{x_0\}$ converging to $x_0$ we have $\liminf\, f (y_k)\geq L$.

This theorem is valid in an arbitrary metric space.

Properties

From Theorem 6 it can be easily concluded that \[ \liminf_{x\to x_0}\,\, f(x) = - \limsup_{x\to x_0}\, (- f(x)) \] \[ \liminf_{x\to x_0}\,\, (\lambda f (x)) = \lambda\, \liminf_{x\to x_0}\,\, f(x) \qquad \limsup_{x\to x_0}\, (\lambda f(x)) = \lambda\, \limsup_{x\to x_0}\, f(x)\qquad \mbox{when '"`UNIQ-MathJax81-QINU`"'} \] and that \[ \liminf_{x\to x_0}\,\, (f(x) + g(x))\geq \liminf_{x\to x_0}\,\, f(x) + \liminf_{x\to x_0}\,\, g(x) \qquad \limsup_{x\to x_0}\, (f(x) + g(x))\leq \limsup_{x\to x_0}\, f(x) + \limsup_{x\to x_0}\, g(x) \] if the additions are not of the type $-\infty + \infty$.

The function $f$ has a finite limit $L$ (resp. has limit $\infty$, $-\infty$) if the lower and upper limits coincide and are equal to $L$ (resp. $\infty$, $-\infty$).

Moreover, we have the following

Proposition Consider the closed set $E'$ of accumulation points of $E$ and define \[ \underline{f} (x) := \liminf_{y\to x}\,\, f(y)\, \] \[ \overline{f} (x) := \limsup_{y\to x}\, f(y)\, . \] $\underline{f}$ is lower semicontinuous and $\overline{f}$ is upper semicontinuous.

From metric spaces to sequences

Consider the space $X=\mathbb N\cup\{\infty\}$ with the metric: \[ d (n,m) = \left|\frac{1}{m+1} - \frac{1}{n+1}\right|,\qquad d(n,\infty) = \frac{1}{n+1}\, . \] Given a sequence of real numbers $\{x_n\}$ consider the function $f:\mathbb N\to \mathbb R$ given by $f(n) = x_n$ as a function defined on a subset of $X$. Then $\limsup_n\, x_n$ and $\liminf_n\,\, x_n$ in the sense of Definition 1 coincide with \[ \limsup_{n\to\infty}\, f(n)\qquad \liminf_{n\to\infty}\,\, f (n) \] in the metric-space sense of Definition 6.

Upper and lower limit of sets

If $\{A_k\}$ is a sequence of subsets of $X$, the upper and lower limit of the sequence $\{A_k\}$ is defined as \[ \limsup_{k\to\infty}\, A_k = \bigcap_{n\in\mathbb N}\, \bigcup_{k\geq n} A_k\, \] \[ \liminf_{k\to\infty}\, A_k = \bigcup_{n\in\mathbb N}\, \bigcap_{k\geq n} A_k\,. \]

References

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