Let $f_n$ be an equicontinuous sequence of functions on a compact interval $D$ and suppose $f_n \to f$ pointwise.
I wrote what I think is a solution, but I never used the assumption that $D$ was compact, so I think it's wrong.
Let $\varepsilon > 0$ be given. Consider an arbitrary $x \in D$ and some $y$ such that $|x - y| < \delta$, where $\delta$ was the same one from equicontinuity definition.
Now, let $N = \sup_{x \in D}\{N_1:\text{pointwise convergence holds $\forall n \geq N_1$}\}$. Then, for all $n > N$, we have that
$$|f_n(x) - f(x)| \leq |f_n(x) - f_n(y)| + |f_n(y) - f(y)| + |f(y) - f(x)| \leq \frac \varepsilon {3} + \frac \varepsilon {3} + \frac \varepsilon {3} = \varepsilon$$
where the first fraction follows from the equicontinuity and the second follows from the pointwise convergence, as $n > N \geq N'$ where $N'$ would have been the $N'$ such that $\forall n \geq N'$, $|f_n(y) - f(y)| < \varepsilon$, and the third term follows if you take limit as $n \to \infty$ of equicontinuity.
Is this right, and if not, where is the mistake?