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Is this true?

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

No — it is false.

The counterexample

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

The kind of mistake this is

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this
uniform convergencecompactness

Others that fail the same way

From Sequences and Series of Functions › Arzelà–Ascoli and equicontinuity

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