Homework problem.
Let $a_1, a_2, ..., a_n$ and $b_1,b_2,...,b_n$ be sets of real numbers. Show that: $ \left(\sum_{k=1}^n a_kb_k\right)^2 \leq \left(\sum_{k=1}^n ka_k^2\right) \left(\sum_{k=1}^n\frac{b_k^2}{k}\right)$
for all $n \geq 1$.
The hint given to us was not to prove this with induction, but to think of the problem "in linear algebra terms".
I've pondered this for a few days now, and come up with this: You can think of the $a$'s as a vector $\langle a_1,...,a_n\rangle$, and the $b$'s as a vector $\langle b_1,...,b_n\rangle$ and then the problem can be rephrased as inner products: $\langle A,B\rangle\langle A,B\rangle \space \leq \space \langle A,A\rangle\langle K^{-1}B,B\rangle\;,$
where $A$ and $B$ are defined above and $KA$ is $\langle 1a_1, 2a_2, ..., na_n\rangle$ and $K^{-1}B$ is $\langle 1b_1, \frac{1}{2}b_2,...,\frac{1}{n}b_n\rangle$.
I'm aware of the similarity with the Cauchy-Schwarz inequality, but can't figure out how to manipulate what I have any further.
Any insights are appreciated.