Real Analysis – Counterexample to Leibniz Criterion for Alternating Series

Question

I have this statement and I need to say if it is true or false:

Let $\{a_n\}$ be a real sequence.

$$\lim_{n\to +\infty} a_n = 0 \quad \implies \quad \sum_{n=1}^{\infty}(-1)^na_n \quad \text{converges}$$

I know, from the Leibniz criterion that:

If $a_n \to 0$, $a_n$ is decreasing and positive then $\sum_{n=1}^{\infty}(-1)^na_n$ converges

From this fact I believe that the statement is false but I couldn't come up with an infinitesimal sequence that isn't decreasing and for that reason is divergent. I tried some function with $sin(\frac{1}{n})$ without any luck.
Any help would be very appreciated, thank you!

Answer 1

HINT:

What happens if $a_n=(-1)^n\frac1n$?