if $\sum_{k=1}^\infty \|x_k\|$ converges , then $\sum_{k=1}^\infty x_k$ also converges.

Question

Let $(x_k)$ be a sequence in a Banach space $X$. if $\sum_{k=1}^\infty \|x_k\|$ converges , then $\sum_{k=1}^\infty x_k$ also converges.

$\textbf{My attempt}$

Let $(e_k)$ be the orthonormal basis of $X$. since $\sum_{k=1}^\infty \|x_k\|$ converges there exists $ 0< M <\infty$ such that $\|x_k\|<M$, then we can write $$|\sum_{k=1}^\infty x_k| \le \sum_{k=1}^\infty |x_k e_k| \le \|x_k\|\|e_k\| < \infty$$

since the series converges absolutely then it converges in ordinary sense.

Answer 1

You cannot write $\sum x_n$ before proving that the series converges. So your attempt fails. Prove that $\left\|\sum\limits_{k=n}^{m} x_n\right\| \to 0$ as $n,m \to \infty$ which makes the partial sum sequence a Cauchy sequence. Then use completeness to finish the proof.