Show that if $f$ is not uniformly continuous on an interval $[a,b]$ then there are sequences $\{x_n\}$ and $\{y_n\}$ chosen from $[a,b]$ so that $x_n-y_n\to0$ but $|f(x_n)-f(y_n)|>c$ for some $c>0$
I've only barely started the proof for this one.
Proof
Suppose $f:[a,b]\to \mathbb{R}$ is not uniformly continuous
Then $\exists c >0: \forall \delta >0,$ there are $x_ ,y \in [a,b]: |x-y|\leq \delta \quad \text{and}\quad |f(x)-f(y)|\geq c$
Let $\{x_n\}, \{y_n\} \subseteq \mathbb{R}$
...
I'm not sure where to go from here. I get stuck because we don't know what $\{x_n\}$ and $\{y_n\}$ converge to, which makes using the definition of convergence tricky here. Any ideas?