Suppose $\left(a_{n}\right)$ is a monotonic sequence of nonnegative real numbers converging to 0. Assume also that the series $\sum\frac{a_{n}}{\sqrt{n}}$ is convergent. Prove that the series $\sum a_{n}^{2}$ is also convergent. Prove that if the monotonicity hypothesis is dropped, then the above conclusion is not true.
If we show that $\sqrt{n}a_{n}\leq B$ for all n for some B , then we have $\sum a_{n}^{2}=\sum\left(\sqrt{n}a_{n}\right)\left(\frac{a_{n}}{\sqrt{n}}\right)\leq B\sum\left(\frac{a_{n}}{\sqrt{n}}\right)<\infty.$
So now I want to show that there exists such a $B$. What makes me think that this must hold is that $\lim\frac{a_{n}}{\sqrt{n}}=0$ and hence $a_{n}$ is approching 0 faster than $\sqrt{n}$ is approaching $\infty$ (or equivalently faster than $\frac{1}{\sqrt{n}}$ is approching 0). But how do I show this? Any ideas?