Category:Uniform Convergence

From ProofWiki
Jump to navigation Jump to search

This category contains results about Uniform Convergence.
Definitions specific to this category can be found in Definitions/Uniform Convergence.


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 $n \to \infty$.