For questions concerning the definition and properties of limit superior and limit inferior of sequences of sets or real numbers.
For questions concerning the definition and properties of limit superior and limit inferior.
The limit superior and the limit inferior of a sequence $a_1,a_2,a_3,\ldots$ of real numbers are real numbers, $+\infty$ or $-\infty$. More precisely, the limit superior is the limit of the sequence $\sup_{n\in\mathbb N}${$a_n,a_{n+1},a_{n+2},\ldots$} and the limit inferior is the limit of the sequence $\inf_{n\in\mathbb N}${$a_n,a_{n+1},a_{n+2},\ldots$}. Each sequence of real numbers has one and only one limit superior and one and only one limit inferior. They are equal if and only if the sequence has a limit.
The limit superior and the limit inferior of a sequence $S_1,S_2,S_3,\ldots$ of sets are respectively the sets $\limsup_nS_n=\bigcap_{i=1}^{+\infty}\bigcup_{j=i}^{+\infty}S_j$ and $\liminf_nS_n=\bigcup_{i=1}^{+\infty}\bigcap_{j=i}^{+\infty}S_j$.
Use this tag along with real-analysis or metric-spaces as is found appropriate. For questions concerning the limsup/inf of sets, please add the elementary-set-theory tag. For questions involving abstract partial orders, use also the order-theory tag.
For questions concerning the evaluation and other properties of limits, use the limit tag.
