Proving that the unit ball in $\ell^2(\mathbb{N})$ is non-compact

real-analysis compactness functional-analysis lp-spaces

A sequence satisfying your criteria cannot have a convergent subsequence, since no subsequence of such a sequence can be Cauchy. But, compact metric spaces are sequentially compact.

To find the sequence $(x_n)$, consider the unit vectors in $\ell_2$ (suitably scaled).

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