[Math] Prove that if $\sum a_n$ converges absolutely, then $\sum a_{2n}$ converges.

Question

In posting this question, I noticed a lot of 'similar' threads pop up, but felt that they required a fundamentally different approach.

If any of you feel differently, please feel free to vote this thread as a duplicate and I will delete it.

Here is my approach, I would appreciate help/correction where relevant, as I am unsure of how robust my answer is:

If $\sum a_n$ converges absolutely, then we have that $\sum |a_n|$ converges.
This implies that $\sum a_n$ also converges, and that the sequence $(a_n)_{n \in \mathbb{N}}\to 0$.$^{(1)}$

This means that the sequence $(a_n)$ is bounded and monotone (decreasing), and as such convergent.

Looking at $(a_{2n})$ we notice it is a sub-sequence of $(a_n)$ and converges to the same value (Bolzano-Weiestrass).$^{(2)}$

Then the partial sums $(s_{2n}) \to A$ and thus $\sum a_{2n} = A$

Is this sufficient? I feel it is a little wishy/washy in using bolzano-weistrass.

Do I need to formally show $(a_{2n})$ is a sub-sequence of $(a_n)$

Any tips/hints/corrections are greatly appreciated.

$^{(1),}$$^{(2)}$ – I am not required to show these results, as quoting the theorem from my notes is considered sufficient by my professor.

Answer 1

$$ \sum_{n\in \mathbb{N}} |a_{2n}| \leq \sum_{n\in \mathbb{N}}|a_n| < \infty$$