A sequence converges iff its liminf and limsup agree

Statement

Let $(a_n)$ be a bounded real sequence. Then $(a_n)$ converges if and only if $\liminf_{n \to \infty} a_n = \limsup_{n \to \infty} a_n$. In this case, $\lim_{n \to \infty} a_n$ is equal to this common value.

Remarks

The proof of the direction from right to left uses $\inf_{k \ge n} a_k \le a_n \le \sup_{k \ge n} a_k$ and the squeeze theorem. In the extended real line, the statement holds for every sequence. Then convergence to $+\infty$ or to $-\infty$ is allowed. For $(-1)^n$, the two values are $-1$ and $1$. They differ, so the sequence diverges.

Explanations

L1 human

The limit inferior and the limit superior are the lowest and the highest points where the sequence keeps returning. If these two points are the same, the sequence has nowhere else to go, so it converges to that point.