\begin{definition}[Definition:Pointwise Convergence]
Let $\sequence {f_n}$ be a sequence of real functions defined on $D \subseteq \R$.
Suppose that:
:$\ds \forall x \in D: \lim_{n \mathop \to \infty} \map {f_n} x = \map f x$
That is:
:$\forall x \in D: \forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n > N: \size {\map {f_n} x - \map f x} < \epsilon$
Then '''$\sequence {f_n}$ converges to $f$ pointwise on $D$ as $n \to \infty$'''.
(See the definition of convergence of a sequence).
\end{definition}