Obscure inequality in a passage of a proof (maximal ergodic theorem)

Question

Look at the following excerpt from the book "Einsiedler and Ward- Ergodic Theory, with a view towards Number Theory":

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I don't understand why the inequality in the red box is valid. Maybe the explanation is simple, but not for me. Can you help me?

Thanks in advance.

Answer 1

You are given that $UF_N +f \geq f_{n+1}$ for all $0\leq n\leq N$, hence $$ UF_N +f \geq \max_{0\leq n\leq N}f_{n+1} = \max_{1\leq n\leq N+1}f_n \geq \max _{1\leq n\leq N}f_n. $$