Limit inferior and limit superior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.

The limit inferior of a sequence ${\displaystyle (x_{n})}$ is denoted by $${\displaystyle \liminf _{n\to \infty }x_{n}\quad {\text{or}}\quad \varliminf _{n\to \infty }x_{n},}$$ and the limit superior of a sequence ${\displaystyle (x_{n})}$ is denoted by $${\displaystyle \limsup _{n\to \infty }x_{n}\quad {\text{or}}\quad \varlimsup _{n\to \infty }x_{n}.}$$

The limit inferior of a sequence (x_n) is defined by $${\displaystyle \liminf _{n\to \infty }x_{n}:=\lim _{n\to \infty }\!{\Big (}\inf _{m\geq n}x_{m}{\Big )}}$$ or $${\displaystyle \liminf _{n\to \infty }x_{n}:=\sup _{n\geq 0}\,\inf _{m\geq n}x_{m}=\sup \,\{\,\inf \,\{\,x_{m}:m\geq n\,\}:n\geq 0\,\}.}$$

Similarly, the limit superior of (x_n) is defined by $${\displaystyle \limsup _{n\to \infty }x_{n}:=\lim _{n\to \infty }\!{\Big (}\sup _{m\geq n}x_{m}{\Big )}}$$ or $${\displaystyle \limsup _{n\to \infty }x_{n}:=\inf _{n\geq 0}\,\sup _{m\geq n}x_{m}=\inf \,\{\,\sup \,\{\,x_{m}:m\geq n\,\}:n\geq 0\,\}.}$$

Alternatively, the notations ${\displaystyle \varliminf _{n\to \infty }x_{n}:=\liminf _{n\to \infty }x_{n}}$ and ${\displaystyle \varlimsup _{n\to \infty }x_{n}:=\limsup _{n\to \infty }x_{n}}$ are sometimes used.

The limits superior and inferior can equivalently be defined using the concept of subsequential limits of the sequence ${\displaystyle (x_{n})}$. An element ${\displaystyle \xi }$ of the extended real numbers ${\displaystyle {\overline {\mathbb {R} }}}$ is a subsequential limit of ${\displaystyle (x_{n})}$ if there exists a strictly increasing sequence of natural numbers ${\displaystyle (n_{k})}$ such that ${\displaystyle \xi =\lim _{k\to \infty }x_{n_{k}}}$. If ${\displaystyle E\subseteq {\overline {\mathbb {R} }}}$ is the set of all subsequential limits of ${\displaystyle (x_{n})}$, then

and

If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i.e. the extended real number line) are complete. More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.