Converg. Seq. of Functions, Derivatives Bounded, Limit not Differentiable

Homework Statement

Find a sequence of differentiable functions $f_n\colon [a,b]\rightarrow\mathbb(R)$ s.t.:
--there exists $M>0$ with $|f_n'(x)|\leq M$ for all $n\in\mathbb{N}$ and $x\in[a,b]$;
--for all $n\in\mathbb{N}$, $|f_n(a)|\leq M$;
--$(g_n)$ is a convergent subsequence with $lim_{n\rightarrow\ifty}g_n(x)=g$ for $f$ NOT DIFFERENTIABLE.

Homework Equations

Arzela-Ascoli for the reals.

The Attempt at a Solution

I have already proved, using Arzela-Ascoli, that such a $g$ exists for any sequence $(f_n)$ fulfilling the first two conditions. But I simply cannot come up with a concrete example where the limit is not differentiable!

Thanks!