Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / P / 3 / e / Solution

The best-approximation characterization gives $⟨f-q,q⟩=0$, since $q\in {\Pi }_k$. ConsequentlyTaking nonnegative square roots proves $\boxed{\Vert q\Vert \le \Vert f\Vert }$. Thus an orthogonal projection cannot increase the norm.