= Solution
A <metric space> is a <complete metric space> when every <Cauchy sequence> converges to a point of that space: for each $\varepsilon>0$, the condition $d(x_m,x_n)<\varepsilon$ for all sufficiently large $m,n$ must imply existence of $x$ with $d(x_n,x)\to0$.
A <Cauchy sequence> in $[0,1]$ is also Cauchy in $\mathbb R$, so it converges there by the assumed completeness. Its limit remains between zero and one, since inequalities are preserved under limits. Therefore
$$
\boxed{[0,1]\text{ with the Euclidean metric is complete}.}
$$
This is an instance of the fact that a <closed subset> of a <complete metric space> is complete with its induced <metric>.
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