Editing Uniform convergence (section)

== Properties ==

* Every uniformly convergent sequence is locally uniformly convergent.
* Every locally uniformly convergent sequence is [[compactly convergent]].
* For [[locally compact space]]s local uniform convergence and compact convergence coincide.
* A sequence of continuous functions on metric spaces, with the image metric space being complete, is uniformly convergent if and only if it is [[Uniformly Cauchy sequence|uniformly Cauchy]].
* If <math>S</math> is a [[compact space|compact]] interval (or in general a compact topological space), and <math> (f_n)</math> is a [[monotonic|monotone increasing]] sequence (meaning <math> f_n(x) \leq f_{n+1}(x)</math> for all ''n'' and ''x'') of ''continuous'' functions with a pointwise limit <math> f</math> which is also continuous, then the convergence is necessarily uniform ([[Dini's theorem]]). Uniform convergence is also guaranteed if <math> S</math> is a compact interval and <math>(f_n)</math> is an [[equicontinuity|equicontinuous]] sequence that converges pointwise.