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I have a linear operator $$A: \ell_1 \rightarrow \ell_1$$ $$A_nx=(0,0,...,0,x_{n+1},x_{n+2},...)$$ with $n$ zeros. I am asked to show two things: that for any $x \in \ell_1$, $\lim_{n \rightarrow \infty}A_nx= 0$, and that the sequence $(A_n)_{n \ge 1}$ in $L(\ell_1)$ doesn't converge. The first part seems self-evident, as I will be left with an infinite number of zeros which is obviously then x=0. But I'm struggling to prove the second part as it seems intuitively not true. Can anyone offer any hints?

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For the first part, notice that if $x\in\ell^1$, then $\sum_{k\geqslant n}|x_k|\to 0$ as $n$ goes to infinity.

For the second part, take $x$ the sequence whose $(n+1)$-th term is $1$ and all the other $0$: we find that the norm of $A_n$ is $1$.

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