Basic properties of lim sup and lim inf

Basic properties of $\limsup$ and $\liminf$: Let $(a_n)$ be a real sequence and define

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 a finite real number, then there exists a subsequence $(a_{n_j})$ with $a_{n_j}\to \ell$ (and similarly for $\liminf$). See limit superior and limit inferior.