It is often useful to take the limit of a sequence of *functions*, with the codomain being a subset of $\mathbb{R}$(or more generally, a metric space). Although it may seem very obvious but there are some subtleties on how to define the convergence of such a sequence. There are two approaches: Pointwise: A sequence of functions $f_{n}$ with $f_{n}:A\to \mathbb{R}$ where $A$ is an arbitrary set, is said to converge pointwise to $f:A\to \mathbb{R}$ iff for every $x$ in $A$ and every $\epsilon>0$ there exists $N \in \mathbb{N}$ such that $n>N$ implies $|f_{n}-f|<\epsilon)$ Uniform: A sequence of A sequence of...