Banach spaces, partial sums converge?

Question

Let $(X,||.||)$ be a Banach space. Suppose the sequence $(x_n) \subset X$ is such that $\sum_{n=1}^{\infty} \|x_n\|$ converges in $\mathbb{R}$.

Prove that $S_n = x_1+x_2+...+x_n$ the "Partial sums" converge in $(X,\|.\|).$

Thanks!

Answer 1

Hint: Show $S_n$ is a Cauchy sequence by estimating $\|S_m - S_n\|$.