Definition:Uniform Convergence/Metric Space

Definition
Let $S$ be a set.

Let $M = \left({A, d}\right)$ be a metric space.

Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: S \to A$.

Let:
 * $\forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N, \forall x \in S: d \left({f_n \left({x}\right), f \left({x}\right)}\right) < \epsilon$

Then $\left \langle {f_n} \right \rangle$ converges to $f$ uniformly on $S$ as $n \to \infty$.

Also defined as
Some sources insist that $N \in \N$ but this is unnecessary and makes proofs more cumbersome.

Also see

 * Definition:Convergent Sequence