Let $(U,V)$ be joint random variables (assume zero means for simplicity). It is well known that \begin{align} \min_{a,b} E[(V -(aU+b))^2], \end{align} is minimized by $a= \frac{E[VU]}{E[U^2]}$ and $b=0$. This answer results in what is know as the best linear estimator.
The classical proof of this expands the expression and finds partial derivatives with respect to $a$ and $b$.
My question: Are there alternative proofs that do not use differentiation? For example, can this be found via some known inequalities like Jensen or Cauchy-Schwarz? In other words, I am looking for interesting and unique solutions to this old problem.
Edit I didn't ask this originality, but can it be also done with minimal expansion.