Recently I am reading a book about ODE, and I find a question that asks to prove that ODE has a periodic solution under some conditions.
Consider the ODE $x'=f(t,x)$, where $x$ is a scalar and $f$ and $\partial f/\partial x$ are continuous in $(t,x)$. Suppose that $f$ is real and of periodic in $t$. Prove that the ODE has a periodic solution if a solution $\psi$ satisfies $\lim_{t\to\infty}\sup |\psi(t)|<\infty$.
I think this is a such useful result, but I am afraid that I don't find the way how to prove this proposition.