Exercise 17.8 [vi-contraction-exercise]

Equation (vi-contraction-equation) on page vi-contraction-equation states that the Bellman operator is a contraction.

1. Show that, for any functions $f$ and $g$, $|\max_a f(a) - \max_a g(a)| \leq \max_a |f(a) - g(a)|\ .$

2. Write out an expression for $|(B\,U_i - B\,U'_i)(s)|$ and then apply the result from (1) to complete the proof that the Bellman operator is a contraction.