Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent

Theorem
Let $S \subseteq \R$.

Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.

Then $\sequence {f_n}$ is uniformly Cauchy on $S$ $\sequence {f_n}$ converges uniformly on $S$.

Sufficient Condition
Let $\sequence {f_n}$ be uniformly Cauchy on $S$.

Necessary Condition
Let $\sequence {f_n}$ converge uniformly on $S$ to $f$.