Proving X Complete: Analysis Homework

[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.