Show that any periodic points of a one-dimensional ODE is a fixed point
My attempt:
If $x_0$ is not a fixed point then there exists some $\epsilon>0$ and $T > 0 $ so that if $|x-x_0|<\epsilon$, we have $|\phi(t,x)-x_0|>\epsilon$ for any $t>T$. But if $x_0$ is a periodic point, then ...
Don't know what else to do. I only know the definitions of fixed points, periodic points, etc.