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 ?

$$\left(\sum|x|\right)^2\geq\sum|x|^2$$ is true for partial sums, so it will be true for the infinite series too. So if $\sum|x|$ is finite, so is$\sum|x|^2$. Your second statement is just the contrapositive of the first, but I think you know that and are just stating it differently.
