### Artificial IntelligenceAIMA Exercises

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.

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.

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