If X is complete, does every absolutely convergent series converge?

[BobSun]

Homework Statement

Let X be a normed linear space. Prove that X is complete if and only if $$\sum^{\infty}_{n=1} x_{n}$$ converges in X for all sequences ($$x_{n}$$) that satisfy $$\sum^{\infty}_{n=1} \left\|x_{n}\right\|$$< $$\infty$$

 

[morphism]

What have you tried? This is actually a familiar fact from calculus. Do you remember how in R a series converged if it converged absolutely? And do you remember how that was proved? Same proof works here.