Pointwise_convergence

Definition

For each $n \in \mathbb{N}$ let $f_{n}$ be a function defined on a set $A \subseteq \mathbb{R}$. The sequence $\left. \left( f_{n} \right) \right.$ of functions converges pointwise on $A$ to a function $f$ if, for all $x \in A$, the sequence of real numbers $f_{n}\left. (x) \right.$ converges to $f\left. (x) \right.$.

In this case, we write $f_{n} \rightarrow f$, $\lim f_{n} = f$, or $\lim\limits_{n \rightarrow \infty}f_{n}\left. (x) \right. = f\left. (x) \right.$. This last expression is helpful if there is any confusion as to whether $x$ or $n$ is the limiting variable.