Composition of a monotonic and periodic function.

Question

We have functions in the domain of the real numbers. $f$ is a monotonic function and $h$ is a periodic function. Show that the composition $f\circ h$ is periodic. I have tried so hard to use the definitions and then come to a conclusion but I cannot solve this. As I understand it, for it to be periodic, there must exist a $T$ (different from $0$) so that this is true: $$f(h(x) + T)=f(h(x)).$$ What can I do now? Thanks in advance for your help.

Answer 1

Let $T$ be the period of $h$. Then $f(h(x+T)) = f(h(x))$, and hence $f \circ h$ is periodic with period $T$.

$f(h(x)+T)$ has nothing to do with this.