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Section 25.4 Interchanging limit processes
Objectives
Determine when limit processes can be interchanged using uniform convergence.
Identify conditions under which limits and derivatives can be interchanged.
Use equicontinuity and the Arzelà-Ascoli theorem to guarantee uniformly convergent subsequences.
Uniform convergence preserves uniform continuity, boundedness, and integrability. Additionally, if the derivatives converge uniformly, then uniform convergence preserves differentiability.
The sequence
\((f_{k})\) is equicontinuous if for all
\(\eps>0\text{,}\) there exists a single
\(\delta>0\) such that
\(|x-y|<\delta\) implies
\(|f_{k}(x)-f_{k}(y)|<\eps\) for all
\(x,y\in A\) and for all
\(k\in\bb N\text{.}\) (It’s like the sequence of functions version of uniform continuity, in a way?)
Arzelà-Ascoli theorem: If
\((f_{k})\) is uniformly bounded and equicontinuous on
\(A\text{,}\) then
\((f_{k})\) contains a uniformly convergent subsequence. (This is like functions version of Bolzano-Weierstrass.)
