Definition:Uniform Convergence/Metric Space

Definition

Let $S$ be a set.

Let $M = \struct {A, d}$ be a metric space.

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

Let:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N, \forall x \in S: \map d {\map {f_n} x, \map f x} < \epsilon$

Then $\sequence {f_n}$ converges to $f$ uniformly on $S$ as $n \to \infty$.

Also defined as

In definining uniform convergence, some sources insist that $N \in \N$, but this is unnecessary and makes proofs more cumbersome.

Also see

• Definition:Convergent Sequence

Sources

• 1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): $\S 9.5$: The Cauchy Condition for Uniform Convergence: Example