$f'_n(x)$ is bounded and $f_n(x) \to 0$ for each x, then $f_n(x) \to 0$ uniformly

Question

I want to show the question

If $f_n(x)$ is differentiable on [a.b] with $|f'_n(x)|<10$ for all n and if $f_n(x) \to 0$ at each x, then $f_n(x) \to 0$ uniformly.

I think I should use triangle inequality with mean value theorem.

In my try, $|f_n(x)|\leq|f_n(x)-f_n(a)|+|f_n(a)|<10(x-a)+\epsilon$

But I can't make $(x-a)$ to be small for arbitrary x.

Help me!

Answer 1

Note $[a,b]$ is compact, hence any open cover has a finite subcover, which means for any given $\epsilon$, you can choose a finite number of $x_i$ such that $[a,b]\subset \cup (x_i-\epsilon,x_i+\epsilon)$. Then choose large enough $N$, such that $|f_n(x_i)|<\epsilon$ for these $i$, $n\ge N$.

Then for any given $x$, using your idea to compare with $x_i$ nearby, which guarantees that $|x-x_i|<\epsilon$.