Editing Arzelà–Ascoli theorem (section)

==Statement and first consequences==
By definition, a sequence <math>\{f_n\}_{n \in \mathbb{N}}</math> of [[continuous function]]s on an interval {{math|''I'' {{=}} [''a'', ''b'']}} is ''uniformly bounded'' if there is a number {{math|''M''}} such that

:<math>\left|f_n(x)\right| \le M</math>

for every function {{math|&thinsp;''f<sub>n</sub>''&thinsp;}} belonging to the sequence, and every {{math|''x'' ∈ [''a'', ''b'']}}.  (Here, {{math|''M''}}  must be independent of {{math|''n''}} and {{math|''x''}}.)

The sequence is said to be ''[[Equicontinuity|uniformly equicontinuous]]'' if, for every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that

:<math>\left|f_n(x)-f_n(y)\right| < \varepsilon</math>

whenever {{math|{{!}}''x'' − ''y''{{!}} < ''δ''&thinsp;}} for all functions {{math|&thinsp;''f<sub>n</sub>''&thinsp;}} in the sequence. (Here, {{math|''δ''}} may depend on {{math|''ε''}}, but not {{math|''x''}}, {{math|''y''}} or {{math|''n''}}.)

One version of the theorem can be stated as follows:

:Consider a [[sequence]] of real-valued continuous functions {{math|{&thinsp;''f<sub>n</sub>''&thinsp;}<sub>''n'' ∈ '''N'''</sub>}} defined on a closed and bounded [[interval (mathematics)|interval]] {{math|[''a'', ''b'']}} of the [[real line]]. If this sequence is [[uniformly bounded]] and uniformly [[equicontinuous]], then there exists a [[subsequence]] {{math|{&thinsp;''f<sub>n<sub>k</sub></sub>''&thinsp;}<sub>''k'' ∈ '''N'''</sub>}} that [[uniform convergence|converges uniformly]]. 
:The converse is also true, in the sense that if every subsequence of {{math|{&thinsp;''f<sub>n</sub>''&thinsp;} }}itself has a uniformly convergent subsequence, then {{math|{&thinsp;''f<sub>n</sub>''&thinsp;} }}is uniformly bounded and equicontinuous.

{{Math proof|drop=hidden|proof= 
The proof is essentially based on a [[diagonalization argument]].  The simplest case is of real-valued functions on a closed and bounded interval:

* Let {{math|''I'' {{=}} [''a'', ''b''] ⊂ '''R'''}} be a closed and bounded interval.  If '''F''' is an infinite set of functions {{math|&thinsp;''f''&thinsp; : ''I'' → '''R'''}} which is uniformly bounded and equicontinuous, then there is a sequence ''f<sub>n</sub>'' of elements of '''F''' such that {{math|''f<sub>n</sub>''}} converges uniformly on {{math|''I''}}.

Fix an enumeration {{math|{''x''<sub>''i''</sub>}<sub>''i'' ∈'''N'''</sub>}} of [[rational numbers]] in {{math|''I''}}.  Since '''F''' is uniformly bounded, the set of points {{math|{''f''(''x''<sub>1</sub>)}<sub>''f''∈'''F'''</sub>}} is bounded, and hence by the [[Bolzano–Weierstrass theorem]], there is a sequence {{math|{''f''<sub>''n''<sub>1</sub></sub>} }} of distinct functions in '''F''' such that {{math|{''f''<sub>''n''<sub>1</sub></sub>(''x''<sub>1</sub>)} }} converges.  Repeating the same argument for the sequence of points {{math|{''f''<sub>''n''<sub>1</sub></sub>(''x''<sub>2</sub>)} }}, there is a subsequence {{math|{''f''<sub>''n''<sub>2</sub></sub>} }} of {{math|{''f''<sub>''n''<sub>1</sub></sub>} }} such that {{math|{''f''<sub>''n''<sub>2</sub></sub>(''x''<sub>2</sub>)} }} converges.

By induction this process can be continued forever, and so there is a chain of subsequences

:<math>\left \{f_{n_1} \right \} \supseteq \left \{f_{n_2} \right \} \supseteq \cdots</math>

such that, for each {{mvar|k}} = 1, 2, 3, ..., the subsequence {{math|{''f<sub>n<sub>k</sub></sub>''} }} converges at {{math|''x''<sub>1</sub>, ..., ''x<sub>k</sub>''}}.  Now form the diagonal subsequence {{math|{''f''} }} whose {{mvar|m}}th term {{mvar|f<sub>m</sub>}} is the {{mvar|m}}th term in the {{mvar|m}}th subsequence {{math|{''f<sub>n<sub>m</sub></sub>''} }}. By construction, {{mvar|f<sub>m</sub>}} converges at every [[rational point]] of {{mvar|I}}.

Therefore, given any {{math|''ε'' > 0}} and rational {{mvar|x<sub>k</sub>}} in {{mvar|I}}, there is an integer {{math|''N'' {{=}} ''N''(''ε'', ''x<sub>k</sub>'')}} such that

:<math>|f_n(x_k) - f_m(x_k)| < \tfrac{\varepsilon}{3}, \qquad n, m \ge N.</math>

Since the family '''F''' is equicontinuous, for this fixed {{mvar|ε}} and for every {{mvar|x}} in {{mvar|I}}, there is an open interval {{math|''U<sub>x</sub>''}} containing {{mvar|x}} such that

:<math>|f(s)-f(t)| < \tfrac{\varepsilon}{3}</math>

for all {{math|''f''&nbsp;∈&nbsp;'''F'''}} and all {{math|''s'',&nbsp;''t''}} in {{mvar|I}} such that {{math|''s'', ''t'' ∈ ''U<sub>x</sub>''}}.

The collection of intervals {{mvar|U<sub>x</sub>}}, {{math|''x''&nbsp;∈ ''I''}},  forms an [[open cover]] of {{mvar|I}}.  Since {{mvar|I}} is closed and bounded, by the [[Heine–Borel theorem]] {{mvar|I}} is [[compact set|compact]], implying that this covering admits a finite subcover {{math|''U''<sub>1</sub>, ..., ''U<sub>J</sub>''}}.  There exists an integer {{mvar|K}} such that each open interval {{mvar|U<sub>j</sub>}}, {{math|1 ≤ ''j'' ≤ ''J''}}, contains a rational {{mvar|x<sub>k</sub>}} with {{math|1 ≤ ''k'' ≤ ''K''}}.  Finally, for any {{math|''t''&nbsp;∈&nbsp;''I''}}, there are {{mvar|j}} and {{mvar|k}} so that {{mvar|t}} and {{mvar|x<sub>k</sub>}} belong to the same interval {{math|''U<sub>j</sub>''}}.  For this choice of {{mvar|k}},

:<math>\begin{align}
\left |f_n(t)-f_m(t) \right| &\le \left|f_n(t) - f_n(x_k) \right| + |f_n(x_k) - f_m(x_k)| + |f_m(x_k) - f_m(t)| \\
&< \tfrac{\varepsilon}{3} + \tfrac{\varepsilon}{3} + \tfrac{\varepsilon}{3}
\end{align}</math>

for all {{math|''n'', ''m'' > ''N'' {{=}} max{''N''(''ε'', ''x''<sub>1</sub>), ..., ''N''(''ε'', ''x''<sub>''K''</sub>)}.}}  Consequently, the sequence {{math|{''f<sub>n</sub>''} }} is [[uniformly Cauchy]], and therefore converges to a continuous function, as claimed.  This completes the proof.
}}

===Immediate examples===

====Differentiable functions====
The hypotheses of the theorem are satisfied by a uniformly bounded sequence {{math|{&thinsp;''f<sub>n</sub>''&thinsp;} }}of [[derivative|differentiable]] functions with uniformly bounded derivatives. Indeed, uniform boundedness of the derivatives implies by the [[mean value theorem]] that for all {{mvar|x}} and {{mvar|y}},

:<math>\left|f_n(x) - f_n(y)\right| \le K |x-y|,</math>

where {{mvar|K}} is the [[supremum]] of the derivatives of functions in the sequence and is independent of {{mvar|n}}.  So, given {{math|''ε'' > 0}}, let {{math|''δ'' {{=}} {{sfrac|''ε''|2''K''}}}} to verify the definition of equicontinuity of the sequence.  This proves the following corollary:

* Let {{math|{''f<sub>n</sub>''} }} be a uniformly bounded sequence of real-valued differentiable functions on {{math|[''a'', ''b'']}} such that the derivatives {{math|{''f<sub>n</sub>''&prime;} }} are uniformly bounded. Then there exists a subsequence {{math|{''f<sub>n<sub>k</sub></sub>''} }} that converges uniformly on {{math|[''a'', ''b'']}}.

If, in addition, the sequence of second derivatives is also uniformly bounded, then the derivatives also converge uniformly (up to a subsequence), and so on. Another generalization holds for [[continuously differentiable function]]s.  Suppose that the functions {{math|&thinsp;''f<sub>n</sub>''&thinsp;}} are continuously differentiable with derivatives {{math|''f<sub>n</sub>''&prime;}}. Suppose that {{math|''f<sub>n</sub>''&prime;}} are uniformly equicontinuous and uniformly bounded, and that the sequence {{math|{&thinsp;''f<sub>n</sub>''&thinsp;},}} is pointwise bounded (or just bounded at a single point).  Then there is a subsequence of the {{math|{&thinsp;''f<sub>n</sub>''&thinsp;} }}converging uniformly to a continuously differentiable function.

The diagonalization argument can also be used to show that a family of infinitely differentiable functions, whose derivatives of each order are uniformly bounded, has a uniformly convergent subsequence, all of whose derivatives are also uniformly convergent. This is particularly important in the theory of distributions.

====Lipschitz and Hölder continuous functions====
The argument given above proves slightly more, specifically

* If {{math|{&thinsp;''f<sub>n</sub>''&thinsp;} }}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|&thinsp;''f<sub>n</sub>''&thinsp;}}, 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|&thinsp;''f''&thinsp;}} on {{math|[''a'', ''b'']}} that is uniformly bounded and satisfies a [[Hölder condition]] of order  {{math|&alpha;}}, {{math|0 < &alpha; ≤ 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}}.