Basic properties of lim sup and lim inf

Basic properties of $\limsup$ and $\liminf$: Let $(a_n)$ be a real sequence and define $s_n=\sup_{k\ge n} a_k,\qquad i_n=\inf_{k\ge n} a_k.$ Then:

• $(s_n)$ is nonincreasing and $(i_n)$ is nondecreasing,
• the limits exist in the extended reals and $\limsup_{n\to\infty} a_n=\lim_{n\to\infty} s_n,\qquad \liminf_{n\to\infty} a_n=\lim_{n\to\infty} i_n,$
• always $\liminf a_n\le \limsup a_n$,
• if $\liminf a_n=\limsup a_n=L\in\mathbb{R}$, then $a_n\to L$,
• if $\ell=limsup a_n$ is finite, then there exists a subsequence $(a_{n_j})$ with $a_{n_j}\to \ell$ (and similarly for liminf ).

These facts package the “eventual upper and lower behavior” of a sequence and are used to analyze oscillation and subsequential limits.

Examples:

• For $a_n=(-1)^n$, $\limsup a_n=1$ and $\liminf a_n=-1$.
• For $a_n=1+\frac{(-1)^n}{n}$, $\limsup a_n=\liminf a_n=1$, hence $a_n\to 1$.