If $\sum \limits_{n=1}^\infty a_n$ diverges, then does $\sum \limits_{n=1}^\infty a_{2n}$ diverge too?

Question

Assume $a_n \geq 0$ and a decreasing sequence and $\lim a_n = 0$

The examples I have in mind are $\sum \limits_{n=1}^\infty \frac{1}{n}$ and $\sum \limits_{n=1}^\infty \log \left ( 1 + \frac{1}{n} \right ) $ both diverge. But what if we extract a subsequence of even numbered indices.

What about for a general subsequence?