Here's a sketch of the proof. Essentially, w_0 is defined as the closest point to u in W.

Let d be the distance from u to W, . We don't yet know that the infimum is attained, but we can choose a sequence (w_n) in W such that as n→∞.

Apply the parallelogram identity , with , to get . Since , the first term on the left side is ≥d^2, and you should be able to deduce that . Thus (w_n) is a Cauchy sequence and (by the completeness of V) it converges to a point , which is the orthogonal projection of u on W.