The question is: $X$ is a normed vector space, prove that $X$ is complete if and only if for any $\{x_n\}$ in $X$, if $\sum_\limits{n=1}^{\infty}||x_n|| < +\infty$, then $\sum_\limits{n=1}^{\infty}x_n$ converges in $X$.
The question is the same of this question, but in that question only one direction is proved.
I don't know how to prove the other direction, i.e. for any $\{x_n\}$ in $X$, if $\sum_\limits{n=1}^{\infty}||x_n|| < +\infty$, then $\sum_\limits{n=1}^{\infty}x_n$ converges in $X$ $\Rightarrow$ $X$ is complete.
I am thinking constructing $y_n=x_{n+1}-x_n$, but it seems not feasible.
