= Infinite-dimensional bounded sequences can lack convergent subsequences
In the <l2 sequence space>, let $e_n$ have value one in coordinate $n$ and zero elsewhere. This <orthonormal sequence> is bounded: $\|e_n\|_2=1$. For distinct indices $n,m$, its terms obey $\|e_n-e_m\|_2=\sqrt2$. No <subsequence> can therefore be a <Cauchy sequence>, so none converges in the <norm>. Thus even a <complete metric space> can have a <bounded sequence> with no <convergent subsequence>. Finite dimension is essential to the usual <Euclidean space> extension, and in a general <metric space> the relevant replacement is <compactness> or <total boundedness> together with <completeness>.
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