prove that, in a Banach space if $\displaystyle Sum||Xn||$ converges then $\displaystyle SumXn$ converges
Show that the sum is Cauchy, i.e. consider $\displaystyle S_k:=\sum_{n=1}^k X_n$, now consider $\displaystyle ||S_k-S_l||$.
Hint: If $\displaystyle \sum_{n=1}^\infty x_n < \infty$ then $\displaystyle \sum_{n=k}^\infty x_n \rightarrow 0$ as k goes to infinity. (A fact that you can prove using the fact that $\displaystyle x_n \rightarrow 0$).
if X is complete and $\displaystyle \sum^{\infty}_{n=1}||X_n||<{\infty}$ then sequence $\displaystyle S_{k}=\sum^{k}_{n=1}X_{n}$ for $\displaystyle k\epsilon\aleph$ is Cauchy because for k>m
$\displaystyle ||S_{k}-S_{m}||\leq\sum^{k}_{n=m+1}||X_{n}||\rightarrow0$ as $\displaystyle m,k\rightarrow0$
therefore , $\displaystyle S=\sum^{\infty}_{n=1}X_{n}= lim_{k\rightarrow\infty}\sum^{k}_{n=1}X_{n}$ exists in X.
is it true?