Let $f_n:\Bbb R\to\Bbb R$ be differentiable for each $n$ and $|f'_n(x)|\le 1$ for $n\in \Bbb N$, $x\in\Bbb R$. If the sequence $\{f_n\}_{n=1}^{\infty}$ converges pointwise to a function $f$, show that $f$ is continuous.
The trouble I met is that I don't know if $$\left|\lim_{n\to\infty}\lim_{x\to c}\frac{f_n(x)-f_n(c)}{x-c}\right|=\left|\lim_{x\to c}\frac{\lim_{n\to\infty}(f_n(x)-f_n(c))}{x-c}\right|$$ so that the derivative of $f$ is less than $1$ and the existence of differentiability of $f$ implies continuity.
Since the inequality is not true in general, can somebody tell me how to connect with $|f'_n(x)|\le1$?
Hint: Let $x,y\in \mathbb{R}$. Use the fact that there exists $c$ such that $f_n(x)-f_n(y)=(x-y)f_n^{\prime}(c)$, hence you have $|f_n(x)-f_n(y)|\leq [x-y|$