If we have a uniform continuous function $f: X \to Y$, then
- $f$ takes a cauchy sequence in $X$ to a cauchy sequence in $Y$.
Now is this statement true: $f$ is uniformly continuous iff, given $\epsilon > 0$, there is an $N > 0$, such that for every $x,y \in I$,where $I$ is an interval ($x \neq y$) we have $$ \Biggl| \frac{f(x)-f(y)}{x-y} \Biggr| > N \ \Longrightarrow |f(x)-f(y)|< \epsilon.$$
If yes, how to prove it?