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

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.

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.