= Solution
\b[Yes.] A <Cauchy sequence> of <continuous functions> in the <supremum norm> is <Cauchy sequence>[Cauchy] in the given <complete> space of <bounded> functions, so it has a <uniform limit> there. A <uniform limit> of <continuous functions> is <continuous>: at a point, use two uniform errors smaller than $\varepsilon/3$ and <continuity> of one sufficiently late term for the remaining error. Thus the limit is in $C([0,1])$, proving <completeness>. Equivalently this is a <closed subspace> of the given <complete> space.
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