Uniform_convergence

Definition

Let $\left. \left( f_{n} \right) \right.$ be a sequence of functions defined on a set $A \subseteq \mathbb{R}$. Then, $\left. \left( f_{n} \right) \right.$ converges uniformly on $A$ to a limit function $f$ defined on $A$ if, for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that $\left| f_{n}\left. (x) \right. - f\left. (x) \right. \right| < \epsilon$ whenever $n \geq N$ and $x \in A$.

Compare this to the definition of pointwise_convergence.

https://math.stackexchange.com/questions/597765/pointwise-vs-uniform-convergence