Is my proof right and enough?: A non-convergent series of non-negative terms diverges to infinity "Proof that a non-convergent series of non-negative terms diverges to infinity"
I just want to be sure that my proof is right and complete. Could someone help me to check it?
My attempt:
Let $\sum_{n=1}^\infty x_n$ be a non-convergent series of non-negative terms and let $M>0$, since $\sum_{n=1}^\infty x_n$ is non convergent, it does not converges to zero, then there exists $N_1 \in \mathbb{N}$ such that:
$$\left| \sum_{m=1}^{N_1} x_n \right| = \sum_{m=1}^{N_1} x_n \geq M$$
Since $x_n \geq 0$ for all $n$ and since the series is non convergent, there must be an infinite amount of terms that are strictly greater than zero (otherwise the series would converge to the finite sum of positive terms). Hence, there exists $N_2 \in \mathbb{N}$ such that:
$$ \sum_{m=1}^n x_n > \sum_{m=1}^{N_1} x_n \geq M$$ for all $n \geq N_2$
Which completes the proof.
 A: "Not convergent to zero" only implies there exists some $M>0$ so that for infinitely many $N$, $$|\sum_{n=1}^Na_n|>M.$$ It isn't necessarily the case that you can choose $M$ arbitrarily large here. What this means is that "not converging to zero" isn't using the full strength of the assumption that the series does not converge.
As a suggestion, try using the Cauchy criterion for convergence to prove the claim instead.
A: Another way of thinking :
A series of non negative reals either convergent or diverges to $+ \infty$.
Let us consider the series of non negative reals $\sum_{n} x_n$ , then the sequence of partial sum $S_n =\sum_{k=1}^{n} x_k$ is an increasing sequence.
If it is bounded  then converges to $\sup\{S_n : n\in \Bbb{N}\}$ otherwise diverges to $+\infty$.


$\sum_{n=1}^\infty x_n$ not convergent implies the sequence $S_n=\sum_{k=1}^{n} x_k$ is not convergent.
Since $(S_n) $ is an increasing sequence (!) and $(S_n) $ doesn't converge implies $(S_n) $ is unbounded above and $(S_n) \to +\infty$ .
Hence  $\sum_{n=1}^\infty x_n $ diverges to $+\infty$

Everything about a series of non negative terms reduced to a increasing sequence whose behavior is very much known.
