Definition:Pointwise Limit

Definition
Let $S$ be a set, and let $\left({f_n}\right)_{n \in \N}$, $f_n: S \to \R$ be a sequence of real-valued functions.

Suppose that for all $s \in S$, the limit:


 * $\displaystyle \lim_{n \to \infty} f_n \left({s}\right)$

exists.

Then the pointwise limit of $\left({f_n}\right)_{n \in \N}$, denoted $\displaystyle \lim_{n \to \infty} f_n: S \to \R$, is defined by (for all $s \in S$):


 * $\displaystyle \left({\lim_{n \to \infty} f_n}\right) \left({s}\right) := \lim_{n \to \infty} f_n \left({s}\right)$

Pointwise limit thence is an instance of a pointwise operation on real-valued functions.

Also see

 * Pointwise Operation on Real-Valued Functions