Assume $f_n$ is a sequence of uniformly continuous functions.
I was given the following criterion to determine if $f_n$ is equicontinuous.
If $\exists t_n,s_n.t_n-s_n \to 0 \land \|f_n(t_n) - f_n(s_n)\| \not\to 0$ then $f_n$ is not equicontinuous.
If $f_n$ is not equicontinuous then there exists $\epsilon_0$ such that $\forall \delta > 0,t,s \in I. |t-s| < \delta \implies |f_n(t)-f_n(s)| \geq \epsilon_0 \forall n$.
So we take $\delta = \frac 1 m$ and we get a sequence $|f_n(t_m)-f_n(s_m)|$. The double index is something we want to get rid of, here is where the hypothesis of uniformly bounded comes into play.
I was told the converse does not hold. So I want a counterexample for:
If $f_n$ is not equicontinuous then $\exists t_n,s_n.t_n-s_n \to 0 \land \|f_n(t_n) - f_n(s_n)\| \not\to 0$
So what I want to prove is that for some $f_n$:
$f_n$ is not equicontinuous and $\forall t_n,s_n.t_n-s_n \to 0 \implies \|f_n(t_n) - f_n(s_n)\| \to 0$