Limit superior (lim sup)

For a real sequence, the limit of the tail suprema, describing the maximal subsequential limit.

Limit superior (lim sup)

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

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

Then the limit superior of $(a_n)$ is

\limsup_{n\to\infty} a_n := \lim_{n\to\infty} s_n,

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

The number $\limsup a_n$ is the largest value that subsequences can “accumulate at” (more precisely, it equals the supremum of all subsequential limits when those limits are taken in $[-\infty,\infty]$ ).

Examples:

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