$f$ is uniformly continuous on $[a,b]$ and that $g$ is uniformly continuous on $g([a,b])$. Then $f\circ g$ is uniformly continuous on $[a,b]$

Question

Suppose that $f$ is uniformly continuous on $[a,b]$ and that $g$ is uniformly continuous on $g([a,b])$ . Then $f\circ g$ is uniformly continuous on $[a,b]$.

Could anyone give me a hint for answering this question?

Answer 1

Consider that you can make $|f(x)-f(y)|$ as small as you wish by choosing $|x-y|$ small enough.

Consider that you can $|g(f(x))-g(f(y))|$ as small as you wish by choosing $|f(x)-f(y)|$ small enough.

Now, given $\epsilon$, how can you pick a $\delta$ small enough, to ensure that $$|x-y|< \delta \implies |g(f(x))-g(f(y))| <\epsilon \quad \text{for any $x,y \in [a,b]$} $$

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Let $\epsilon$ be given. Since $g$ is uniformely continuous on $f([a,b])$, I can find some $\delta_1$ such that $$|f(x)-f(y)| < \delta_1 \implies |g(f(x))-g(f(y))| <\epsilon$$.

If only I could find some $\delta_2$ such that $|f(x)-f(y)| < \delta_1$...

Now read the first sentence of this post.