A Normed Vector Space is Complete iff all Absolutely Convergent Series Converge
# Statement
Let $(X, ||\cdot||)$ be a Normed Vector Space. Then $X$ is a Complete Metric Space If and Only If for all $({x}{n}){n=1}^{\infty} \subset X$ such that $\sum\limits_{n=1 }^{\infty} ||x_{n}||$ converges, $\sum\limits_{n=1}^{\infty} x_{n}$ converges in $X$.