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I have been watching real analysis lectures on YouTube by Prof. Francis Su and he provided a simpler definition of $\lim \sup$ as

Let $\{s_n\}$ be a sequence of real numbers, then

$\lim\sup s_{n} = \displaystyle\lim_{n \to \infty}\left(\sup_{k \gt n}s_k\right)\tag1$ where $n,k \in \mathbb{N}.$

In Walter Rudin's Principles of Mathematical Analysis the definition for $\lim \sup$ given is as

Let $\{s_n\}$ be a sequence of real numbers. Let E be the set of numbers $x$ (in the extended real number system) such that $\{s_{n_k}\} \to x$ for some subsequence $\{s_{n_k}\}$. This set E contains all subsequential limits, plus possibly the numbers $+\infty, -\infty$ We now put

$s^* = \sup E$

The number $s^*$ is called the upper limit of $\{s_n\}$ and we use the notation

${\displaystyle\lim \displaystyle\sup}_{n \to \infty} s_n = s^*.$

Also some answers here also said that $\lim \sup$ is the infimum of all supremums of $\{s_n\}.$

I know these definitions are equivalent but what is the intuition behind these definitions being same? Like how $\lim\sup s_{n}$ in $(1)$ will always be a subsequential limit and that too supremum of the all subsequtial limits of $s_{n}$?

Soham Gadhave
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  • Note that as a function of $n$, the sequence $(\sup_{k > n}s_k)$ is monotonically decreasing. Hence its infimum equals its limit, assume we are working in the extended reals, to allow for the possibility that the limit is $-\infty$. So that shows that the definitions $\lim_{n \to \infty}\sup_{k > n}s_k$ and $\inf_{n \to \infty}\sup_{k > n}s_k$ are equivalent. The link given by @Crostul shows that Rudin's definition is indeed equivalent to these. –  Sep 09 '19 at 18:49

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