Interchange of limit and sum [closed]

Let $X:=(X(i))\in\ell^1(\mathbb{N})$ and $\{X_n\}$ be sequence in $\ell^1(\mathbb{N})$ such that

(i) $X_n$ converges to $X$ in sup norm,

(ii) $\sum_{i}X_n(i)$ is a Cauchy sequence.

Then is it true that $\sum_{i}X_n(i)$ converges to $\sum_{i}X(i)$?
That is, $$\lim_{n\to\infty}\sum_{i}X_n(i)=\sum_{i}\lim_{n\to\infty}X_n(i)?$$