prove that : $ \sum_{n=0}^\infty |x_n|^2 = +\infty \Rightarrow \sum_{n=0}^\infty |x_n| = +\infty $ i wanted to prove initially that a function is well defined and i concluded that it's enough to prove this statement for $x_n$ a sequence : 
$ \sum_{n=0}^\infty |x_n|^2 = +\infty \Rightarrow \sum_{n=0}^\infty |x_n| = +\infty  $
or :
$ \sum_{n=0}^\infty |x_n| < +\infty \Rightarrow \sum_{n=0}^\infty |x_n|^2 < +\infty  $
any help ?

Hint: If $\sum a_n < \infty$, then $\displaystyle \lim_{n \to \infty} a_n = 0$.  In particular for large $n$, we have $|a_n| \leq 1$.  Then
$$
|a_n|^2 = |a_n| |a_n| \leq \dots
$$
