\begin{definition}[Definition:Pointwise Limit]
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:
:$\ds \lim_{n \mathop \to \infty} \map {f_n} s$
exists.
{{definition wanted|Incorporate infinite limits}}
Then the '''pointwise limit of $\sequence {f_n}_{n \mathop \in \N}$''', denoted $\ds \lim_{n \mathop \to \infty} f_n: S \to \R$, is defined as:
:$\forall s \in S: \ds \map {\paren {\lim_{n \mathop \to \infty} f_n} } s := \lim_{n \mathop \to \infty} \map {f_n} s$
'''Pointwise limit''' thence is an instance of a pointwise operation on real-valued functions.
{{definition wanted|of course, makes sense in arbitrary metric/topological space. cover this}}
\end{definition}