Proof for sequential criteria of non-uniform continuity.

Question

A function $f:A\rightarrow \mathbb R$ is non-uniformly continuous if there exist two sequence $x_n, y_n$ and a particular $\epsilon_0 > 0$ such that $|x_n-y_n|\rightarrow 0$ and $|f(x_n)-f(y_n)|\ge \epsilon_0$.

How do I begin the proof.

Answer 1

It's tautological: if that condition holds, then appropriately large values of $n$ provide counterexamples for the case $\varepsilon=\epsilon_0/2$ in the definition of UC.

Answer 2

Note that, for such a $\varepsilon_0$, you know that, if $f$ was uniformly continuous, there would be a $\delta>0$ such that$$|x-y|<\delta\implies\bigl|f(x)-f(y)\bigr|<\varepsilon_0.$$