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## Homework Statement

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