Let $f$ be an unbounded, non-decreasing function, With the following property:

For any positive sequence $a\_n$ s.t. $\sum a\_n$ converges, $\sum\frac{1}{f(\frac{1}{a\_n})}$ also converges.

Prove, Or disprove:

$\displaystyle\sum \frac{1}{(f(n))^2}$ also converges.

So far, I have tried to show convergence of $\sum \frac{1}{(f(n))^2}$ by showing the limit of $\frac{f(n)^2}{f(1/a\_n)}$ exists but it got me no where and after a couple of attempts I realized that's not the way to go.

Any attempts to find an upper bound failed as well.

Any hints will be appreciated.