Limit superior, limit inferior

Some sequences of practical importance (in science and engineering) may not have limits but have limits superior or limits inferior. While converging sequences are bounded, bounded sequences are not always converging. However, bounded sequences always have limits superior and limits inferior.

First, let us review some basic notions.

Definition (Upper bound, lower bound)

Let $S$ be a subset of $\mathbb{R}$: $S\subset \mathbb{R}$. Let $\alpha \in \mathbb{R}$ be a real number

1. If we have $x\le \alpha$ for all $x\in S$, then $\alpha$ is said to be an upper bound of $S$. 
2. If we have $x\ge \alpha$ for all $x\in S$, then $\alpha$ is said to be a lower bound of $S$. 

Definition (Supremum, infimum)

Let $S$ be a subset of $\mathbb{R}$: $S\subset \mathbb{R}$.

1. If $S$ is bounded above, and there exists the lowest value of the upper bounds, then the lowest upper bound is called the supremum of $S$, denoted $supS$. That is, if $U(S)$ represents the set of all upper bounds of $S$, then $$supS=minU(S).$$
2. If $S$ is bounded below, and there exists the largest value of the lower bounds, then the largest lower bound is called the infimum of $S$, denoted $infS$. That is, if $L(S)$ represents the set of all lower bounds of $S$, then $$infS=maxL(S).$$

Let $\{a_n\}=\{a_1,a_2,\cdots \}$ be a bounded sequence. For any $n\in \mathbb{N}$, we define ${\bar{a}}_n$ and ${\underset{―}{a}}_n$ by

$$\begin{array}{rcl}{\bar{a}}_n& =& \underset{k\ge n}{sup}{a}_k=sup\{a_k\mid k\ge n\},\\ {\underset{―}{a}}_n& =& \underset{k\ge n}{inf}{a}_k=inf\{a_k\mid k\ge n\}.\end{array}$$

That is, ${\bar{a}}_n$ and ${\underset{―}{a}}_n$ are the supremum and infimum, respectively, of the subsequence $a_n,a_{n+1},a_{n+2},\cdots$. Clearly, we have

$${\underset{―}{a}}_n\le a_n\le {\bar{a}}_n$$

for all $n\in \mathbb{N}$. Noting that

$$\{a_k\mid k\ge n\}\supset \{a_k\mid k\ge n+1\},$$

we have

$$\begin{array}{rcl}{\bar{a}}_n& \ge & {\bar{a}}_{n+1},\\ {\underset{―}{a}}_n& \le & {\underset{―}{a}}_{n+1}.\end{array}$$

Thus, $\{{\bar{a}}_n\}$ and $\{{\underset{―}{a}}_n\}$ are monotone decreasing and increasing, respectively, and we have the following inequalities:

$${\underset{―}{a}}_1\le {\underset{―}{a}}_2\le \cdots \le {\underset{―}{a}}_n\le \cdots \le {\bar{a}}_n\le \cdots \le {\bar{a}}_2\le {\bar{a}}_1.$$

Therefore,

1. The sequence $\{{\bar{a}}_n\}$ is monotone decreasing and bounded.
2. The sequence $\{{\underset{―}{a}}_n\}$ is monotone increasing and bounded.

Recall that any bounded monotone sequence converges. Thus, the following notions of limit superior and limit inferior are well-defined.

Definition (Limit superior and limit inferior)

Let $\{a_n\}$ be a bounded sequence. We define the limit superior of $\{a_n\}$ by

$$\underset{n\to \mathrm{\infty }}{lim sup}{a}_n=\underset{n\to \mathrm{\infty }}{lim}\underset{k\ge n}{sup}{a}_k,$$

and the limit inferior by

$$\underset{n\to \mathrm{\infty }}{lim inf}{a}_n=\underset{n\to \mathrm{\infty }}{lim}\underset{k\ge n}{inf}{a}_k.$$

Remark. Using ${\bar{a}}_n=\underset{k\ge n}{sup}{a}_n$ and ${\underset{―}{a}}_n=\underset{k\ge n}{inf}{a}_n$, we can also write

$$\begin{array}{rcl}\underset{n\to \mathrm{\infty }}{lim sup}{a}_n& =& \underset{n\to \mathrm{\infty }}{lim}{\bar{a}}_n,\\ \underset{n\to \mathrm{\infty }}{lim inf}{a}_n& =& \underset{n\to \mathrm{\infty }}{lim}{\underset{―}{a}}_n.\end{array}$$

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Remark. Some authors prefer to denote the limit superior and limit inferior as

$$\underset{n\to \mathrm{\infty }}{\bar{\lim }}{a}_n$$

and

$$\underset{n\to \mathrm{\infty }}{\underset{―}{\lim }}{a}_n,$$

respectively. □

Example. Let's see a few examples.

1. If $a_n=(-1{)}^{n+1}$, then $\underset{n\to \mathrm{\infty }}{lim sup}{a}_n=1$ and $\underset{n\to \mathrm{\infty }}{lim inf}{a}_n=-1$.
2. If $a_n=\frac{(-1{)}^n}{n}$, then $\underset{n\to \mathrm{\infty }}{lim sup}{a}_n=\underset{n\to \mathrm{\infty }}{lim inf}{a}_n=0$.
3. If $\{a_n\}$ is bounded and monotone increasing with $\alpha =\underset{n\in \mathbb{N}}{sup}{a}_n$, then $\underset{n\to \mathrm{\infty }}{lim sup}{a}_n=\underset{n\to \mathrm{\infty }}{lim inf}{a}_n=\alpha$.

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See also: Limit inferior and limit superior (Wikipedia).