The Projection Theorem

Proof. (Projection Theorem) Showing that the existence of minimizer implies that V is closed is left as an exercise. So we assume that V is closed. For f ∈ H, let α := infv∈V ‖v − f‖. It is a little easier to work with this in an equivalent form, α2 = infv∈V ‖v − f‖2. Thus, for every ε > 0 there is a vε ∈ V such that α2 ≤ ‖vε − f‖2 < α2 + ε. By choosing ε = 1/n, where n is a positive integer, we can find a sequence {vn}n=1 in V such that 0 ≤ ‖vn − f‖ − α < 1 n (1.1)