Limit inferior

Definition

Let $(a_n)$ be a bounded sequence of real numbers. For each $n \in \mathbb{N}$, let $i_n = \inf \{a_k : k \ge n\}$. The sequence $(i_n)$ is increasing and bounded above. The limit inferior of $(a_n)$ is $$\liminf_{n \to \infty} a_n = \lim_{n \to \infty} i_n = \sup_{n \in \mathbb{N}} \inf_{k \ge n} a_k.$$

Remarks

The limit exists by the monotone convergence theorem. It satisfies $\liminf a_n = -\limsup (-a_n)$, where $\limsup$ is the limit superior. Always $\inf_n a_n \le \liminf a_n \le \limsup a_n \le \sup_n a_n$. For $a_n = (-1)^n$, the limit inferior is $-1$ and the limit superior is $1$.

Explanations

L1 human

The limit inferior is the lowest value that the sequence keeps coming back to, in the long run. Each number below it is undercut by only finitely many terms.