Uniform Convergence

Last updated Nov 1, 2022

Let $(M_{i}, d_{i}){i \in I}$ be Metric Spaces with Index Set $I$ and let $M = \prod\limits{i \in I}^{} M_{i}$. Suppose $({f}{n}){n=1}^{\infty} \subset M$. We say $f_{n}$ converges uniformly to $f \in M$ if $\forall \epsilon > 0$, there exists an $N \in \mathbb{N}$ so that for all $n \geq N$ , $|f_{n}(i) - f(i)| < \epsilon$ for all $i \in I$.