Change of scale of periodic function

Question

Could someone show how to prove the following:

If $f(x)$ has a period of $p$ show that $f(kx)$ has period of $p/k$.

Answer 1

$$ f(x)=f(x+p) \implies f(kx) = f(kx+p) = f(k(x+p/k)) $$

so if $g(x) \equiv f(kx)$ then $g(x)=g(x+p/k)$