Sequence $(f_n) \to f_n$ on $S \subseteq \mathbb{R}$ converges uniformly iff $\lim_{n \to \infty} \sup \{f(x)-f_n(x)| : x\in S\} = 0$

Question

I am having trouble proving the converse. Namely, if

$$\lim_{n \to \infty} \sup \{|f(x) - f_n(x)| : x \in S \subseteq \mathbb{R}\} = 0$$

then the sequence $(f_n)$ converges uniformly to $f$

Any hints are appreciated.

Answer 1

Fix $\epsilon\geq 0$. Then there exists $n_0\in \mathbb{N}$ such that $$\sup \{|f(x) - f_n(x)| : x \in S \subseteq \mathbb{R}\}\leq \epsilon \text{ for all } n\geq n_0.$$ Which implies, $$|f(x) - f_n(x)|\leq \epsilon\text{ for all } n\geq n_0, \text{ for any } x\in S.$$ Hence, $f_n\rightarrow f $ as $n\rightarrow\infty$ uniformly.