Generalize the preceding exercise to any Hilbert space. Specifically, let S be a nonempty, closed, convex set in Hilbert space X and y B S. Let

Then there exists a sequence (xn) in S such that || xn - y || †’ d. Show that

1. (xn) is a Cauchy sequence.
2. There exists a unique point x0 e S which is closest to y, that is,
||x0 - y|| ‰¤ ||x - y|| for every x ˆŠ S
To complete this section, we give two important applications of exercise 3.72. Exercise 3.74 was used in chapter 2 to prove Brouwer's fixed point theorem (theorem 2.6).

d = inf llx-yll rES Xg Figure 3.4 Minimum distance to a closed convex set