Limit superior and limit inferior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (i.e., 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, 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 (x_n) is defined by

or

Similarly, the limit superior of (x_n) is defined by

or

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.

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