# Definition:Pointwise Limit

## Definition

Let $S$ be a set.

Let $\sequence {f_n}_{n \mathop \in \N}$, $f_n: S \to \R$ be a sequence of real-valued functions.

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

$\displaystyle \lim_{n \mathop \to \infty} \map {f_n} s$

exists.

Then the pointwise limit of $\sequence {f_n}_{n \mathop \in \N}$, denoted $\displaystyle \lim_{n \mathop \to \infty} f_n: S \to \R$, is defined as:

$\forall s \in S: \displaystyle \map {\paren {\lim_{n \mathop \to \infty} f_n} } s := \lim_{n \mathop \to \infty} \map {f_n} s$

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