Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Necessary Condition

Theorem
Let $S \subseteq \R$.

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

Let $\sequence {f_n}$ be uniformly convergent on $S$.

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

Proof
Fix some $\varepsilon \in \R_{> 0}$.

Since $f_n \to f$ uniformly, there exists some $N \in \N$ such that:


 * $\size {\map {f_n} x - \map f x} < \dfrac \varepsilon 2$

for all $x \in S$ and $n > N$.

Then if $x \in S$ and $n, m > N$, we have:

Since $\varepsilon$ was arbitrary, $\sequence {f_n}$ is uniformly Cauchy on $S$.