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Counterexample

Is this true?

A uniformly bounded sequence in C[0,1] has a uniformly convergent subsequence

No — it is false.

The counterexample

fn(x)=xnf_{n}(x) = x^{n}

Why it fails

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

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Hypothesis dropped

A theorem applied without checking one of its preconditions.

See real exam options built on this mistakeDrill statements like this
uniform convergencecompactness

Go deeper· Explained in full in Arzelà–Ascoli and equicontinuity· From Sequences and Series of Functions › Arzelà–Ascoli and equicontinuity

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