Theorem. Given a nonempty closed convex subset $K$ of a Hilbert space, there exists a point in $K$ of mimimal norm.
Most of the usual proof (Check for example Lax, Functional analysis, 6.2 Theorem 2) is pretty intuitive: we construct a sequence of elements in $K$ whose norms tend to $0$ and then prove that this sequence is Cauchy. It is also easy to find a counterexample for non-convex sets.
One of the steps in the usual proof is applying the parallelogram law. This is very unintuitive and seems to come out of nowhere. Indeed every other step in the proof is very natural. Is there a more natural proof of this, or an intuitive explanation for this step?