How to show sequnce of function converges uniformly on any set on which function is bounded?

Question

I am reading Folland real analysis chapter 2.

In that I come across theorem 2.10

I understand first part about point wise convergence .But How author claim about uniform convergence I do not have idea.

I would be thankful if someone kindly help me.

Any help will be appreciated

Answer 1

Let $U$ be some set on which $f$ is bounded. Say $0 \leq f(x) \leq C$, for all $x \in U$. Fix $N$ such that $C \leq 2^N$. Then for $n \geq N$, $$0 \leq f(x) - \phi_n(x) \leq 2^{-n}$$ for any $x \in U$ by the text since $U \subseteq \{x : f(x) \leq 2^n\}$. This implies that $\phi_n \to f$ uniformly on $U$.