real analysis - Proof writing: $sum_{n=1}^{infty}| a_n|

Let $\sum_{n=1}^{\infty} a_n $ be an absolutely converging series. By definition, this means $\sum_{n=1}^{\infty} \lvert a_n\rvert $ converges. We want to show that $\sum_{n=1}^{\infty} a^2_n $ converges absolutely as well.

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Here is what I tried:

Let's say $\sum_{n=1}^{\infty} \lvert a_n^2\rvert $ converges. Let $\epsilon > 0$ be given and there exists $N \in \mathbb{N} $ such that

$\lvert a^2_{m+1}\rvert+ \lvert a^2_{m+2}\rvert + \dots+ \lvert a^2_{n}\rvert< \epsilon$ for all $ n > m \geq N $. By the triangle inequality,

$$\lvert a^2_{m+1} + a^2_{m+2} + \dots + a^2_{n}\rvert \leq \lvert a^2_{m+1}\rvert+ \lvert a^2_{m+2}\rvert + \dots + \lvert a^2_{n}\rvert $$

so the Cauchy Criterion guarantees that $\sum_{n=1}^{\infty} a^2_n $ also converges.

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Here is my question:

Is this correct, or do I need to prove anything else?