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Arzelà–Ascoli theorem
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====Lipschitz and Hölder continuous functions==== The argument given above proves slightly more, specifically * If {{math|{ ''f<sub>n</sub>'' } }}is a uniformly bounded sequence of real valued functions on {{math|[''a'', ''b'']}} such that each ''f<sub>n</sub>'' is [[Lipschitz continuous]] with the same Lipschitz constant {{mvar|K}}: ::<math>\left|f_n(x) - f_n(y)\right| \le K|x-y|</math> :for all {{math|''x'', ''y'' ∈ [''a'', ''b'']}} and all {{math| ''f<sub>n</sub>'' }}, then there is a subsequence that converges uniformly on {{math|[''a'', ''b'']}}. The limit function is also Lipschitz continuous with the same value {{mvar|K}} for the Lipschitz constant. A slight refinement is * A set {{math|'''F'''}} of functions {{math| ''f'' }} on {{math|[''a'', ''b'']}} that is uniformly bounded and satisfies a [[Hölder condition]] of order {{math|α}}, {{math|0 < α ≤ 1}}, with a fixed constant {{mvar|M}}, ::<math>\left|f(x) - f(y)\right| \le M \, |x - y|^\alpha, \qquad x, y \in [a, b]</math> :is relatively compact in {{math|C([''a'', ''b''])}}. In particular, the unit ball of the [[Hölder condition|Hölder space]] {{math|C<sup>0,''α''</sup>([''a'', ''b''])}} is compact in {{math|C([''a'', ''b''])}}. This holds more generally for scalar functions on a compact metric space {{mvar|X}} satisfying a Hölder condition with respect to the metric on {{mvar|X}}.
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