Let $\{a_i\}, i=1, 2, \cdots$ be a nonincreasing sequence of positive numbers, and suppose $\sum_{k=1}^{\infty}\frac{a_k}{\sqrt{k}}<\infty~$, show that $\sum_{k=1}^{\infty}a_k^2<\infty.$
Could you give a direct proof by finding an inequality of the type $\sum_{k=1}^{\infty}a_k^2\leq f<\infty$, where $f$ is some expression related to the already known number $\sum_{k=1}^{\infty}\frac{a_k}{\sqrt{k}}$?
Thanks in advance!
Remark:
I have solved this problem, but my method is indirect, I have used the uniformly bounded principal and the inequality $lim_{k\to\infty}\frac{1}{\sqrt{k}}\sum_{j=1}^ka_j=0,\forall (a_k)\in l^2$, so I really want to know how to solve the problem by finding a direct inequality.