Limit inferior (lim inf)

For a real sequence, the limit of the tail infima, describing the minimal subsequential limit.

Limit inferior (lim inf)

Let $(a_n)$ be a sequence in the extended real line $[-\infty,\infty]$ . Define the tail infima

i_n := \inf\{a_k : k\ge n\}\in[-\infty,\infty].

Then the limit inferior of $(a_n)$ is

\liminf_{n\to\infty} a_n := \lim_{n\to\infty} i_n,

where the limit exists in $[-\infty,\infty]$ because $(i_n)$ is nondecreasing.

The number $\liminf a_n$ is the smallest value that subsequences can “accumulate at” (it equals the infimum of all subsequential limits in $[-\infty,\infty]$ ).

Examples:

If $a_n=(-1)^n$ , then $\liminf a_n = -1$ .
If $a_n=1/n$ , then $\liminf a_n = 0$ .
If $a_n=-n$ , then $\liminf a_n = -\infty$ .