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(See title.) I think that I know how to prove $\implies$: If $(x_n)$ is a bounded sequence in $\ell^p$, then there is a $c\in\mathbb{R}$ such that $\|x_n\|\leq c$ for all $n$. So $$\|T_a x_n\|=\|a_n x_n\|\leq c\|a_n\|\to0,$$ from which it follows that any subsequence of $(T_a x_n)$ in $\ell^p$ converges (to $0$). (Is this correct?) However, for the converse, I have no idea. Any suggestions are greatly appreciated!

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  • $\begingroup$ See this. $\endgroup$ – David Mitra Nov 9 '19 at 12:03
  • $\begingroup$ I don't see why the identity $\lVert (a_kx_{n,k}\,:\, k\in\Bbb N)\rVert=\lVert a_nx_n\rVert$ should hold. $\endgroup$ – Gae. S. Nov 9 '19 at 12:04
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    $\begingroup$ Perhaps I have overlooked something, but it seems the proposed target duplicate has only one answer, consisting of a "hint" for the direction of implication that OP here says "that I know how to prove". So while both Questions pose an "iff" claim, the direction $\impliedby$ remains open. $\endgroup$ – hardmath Nov 9 '19 at 23:04