I have the following exercise:
Suppose $f\in C^1(\mathbb{R}\times\mathbb{R},\mathbb{R})$ is a function $\omega$ periodic in the first variable. Also there exists $a,b\in\mathbb{R}$ such as $f(t,b)<0<f(t,a)$ $\forall t\in \mathbb{R}$. Prove that for a solution $\phi$ of the DE $y'=f(t,y)$ with initial value $\phi(0)=y_0$, then for $y_0\in[a,b]$ we have $\phi(t,y_0)\in[a,b]$ $\forall t\geq0$
I know that this is related with the Poincare map (PM), also a map $f$ such as $f([a,b])\subset[a,b]$ has a fixed point, so this will allow me to prove that if $y_0$ is a fixed point of the PM, then the ivp with $x(0)=y_0$ has a periodic solution. But I cannot see how to use this hypothesis $f(t,b)<0<f(t,a)$ $\forall t\in \mathbb{R}$ to prove the last statement.