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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 ?

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If $|x_n|\ge 1$ for infinitely many $n$, then $\sum_{n=1}^\infty|x_n|$ diverges.

If $|x_n|\lt 1$ for all large enough $n$, then for large enough $n$ we have $|x_n|\gt |x_n|^2$, so by Comparison $\sum_{n=1}^\infty |x_n|$ diverges.

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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

$$ |a_n|^2 = |a_n| |a_n| \leq \dots $$

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$$\left(\sum|x|\right)^2\geq\sum|x|^2$$ 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.

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