The bookUnit 1 · Sequences and Series of Functions11 / 83

Arzelà–Ascoli and equicontinuity

Why this is asked: Arzelà–Ascoli = uniformly bounded + equicontinuous ⇒ a uniformly convergent subsequence. It is the compactness criterion in C[a,b]; the failures are xⁿ (not equicontinuous) and constants n (not bounded).

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Checking Arzelà–Ascoli: the hypothesis that usually failsinteractive

Two families that look harmless and are not equicontinuous — and exactly where each breaks.

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The trap here

“A uniformly bounded sequence in C[0,1] has a uniformly convergent subsequence” — false

$f_{n}(x) = x^{n}$

Bounded by 1, but the pointwise limit is discontinuous so no subsequence converges uniformly. Equicontinuity is the missing hypothesis.