Editing Arzelà–Ascoli theorem (section)

====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.