Statement: Let $X$ be a Banach space If $x_n \rightarrow x$ weakly and every subsequence of $\{x_n\}$ has a strongly convergent subsequence, then $x_n\rightarrow x$ strongly in $X$
Attempt: ?
2026-03-29 23:41:58.1774827718
weakly convergent subsequence implies strongly convergent
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Suppose $\lim_{n \to \infty}x_n\neq x$. Then there is an $\epsilon \gt0$ and a subsequence $(x_{n_k})$ such that $\|x_{n_k}-x\|\ge\epsilon$ for all $k\in\mathbb{N}$. Use this to derive a contradiction.