TITLE: Convergence of Cauchy sequences in $(\ell^1, \|\cdot\|_\infty)$
QUESTION [5 upvotes]: Do Cauchy sequences in $(\ell^1, \|\cdot\|_\infty)$ converge in $\ell^1$?

I feel that the answer is No, but I am not able to find a counterexample. If $x^{(n)}$ is a sequence in $\ell^1$, then we have $$\sum_{i=1}^\infty |x^{(n)}_i| < \infty$$ for every $n\in\mathbb N$. Also if $x^{(n)}$ is Cauchy w.r.t the sup-norm, then
$$\forall\epsilon>0\exists N\in\mathbb N\forall m,n>N (\|x^{(m)} - x^{(n)}\|_\infty < \epsilon)$$
That is the definition, and that is all I can see. Any ideas?

REPLY [5 votes]: Let us take $x^{(n)} \in \ell^1$ such that
$$
x^{(n)}(k) = 
\begin{cases}
1/k & \text{if } k \le n,\\
0 & \text{otherwise}.
\end{cases}
$$
For $m \ge n+1$,
$$\|x^{(m)} - x^{(n)}\|_\infty  = \frac{1}{n + 1} \to 0$$
but $x^{(n)}$ does not converges in $\ell^1$ because
$$\sum_{k \in \mathbb N} \frac{1}{k} = +\infty.$$