Limit superior

Definition

Let $(a_n)$ be a bounded sequence of real numbers. For each $n \in \mathbb{N}$, let $s_n = \sup \{a_k : k \ge n\}$. The sequence $(s_n)$ is decreasing and bounded below. The limit superior of $(a_n)$ is $$\limsup_{n \to \infty} a_n = \lim_{n \to \infty} s_n = \inf_{n \in \mathbb{N}} \sup_{k \ge n} a_k.$$

Remarks

The limit exists because a decreasing sequence bounded below converges. For a sequence that is not bounded above, one sets $\limsup a_n = +\infty$. For a sequence that is bounded above and not below, the same formula is used in the extended real line, and it can give $-\infty$. For $a_n = (-1)^n (1 + 1/(n+1))$, the limit superior is $1$, although no term equals $1$. For a bounded sequence, it is the largest cluster point.

Explanations

L1 human

The limit superior is the highest value that the sequence keeps coming back to, in the long run. Each number above it is exceeded by only finitely many terms.