what $a_i$ ensures that $\sum^{\infty}_{i=0}a_i^2 < \infty$?

What requirements are needed on $a_i$, a non-negative sequence to ensure that $\sum^{\infty}_{i=0}a_i^2 < \infty$?

My answer:

If the series converges then it must be that $a_i^2$ approaches $0$, so $a_i$ also approaches $0$. So, $a_i$ approaches $0$ and must be bounded.

But, being bounded and $a_i$ approaching $0$ does not guarantee that the series will converge. For example, take $a_i = 1/\sqrt{i}$, then $a_i$ approaches $0$ and its bounded, but the series will diverge since $a_i^2 = 1/i$.

Question: What condition should be imposed then on $a_i$ to ensure that the series of $a_i^2$ will always diverge.? Is there any iff statement that would be applicable?