Let $f:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be a continuous function such that $\frac{df}{dx}$ is well-defined and continuous on $\mathbb{R} \times \mathbb{R}$. Assume $\gamma \in C^1([0,+\infty[)$ is increasing with $\gamma(0) > 0$ and such that $\forall t \ge 0. f(t,\gamma(t)) \le 0 \le f(t,0)$.
Now consider the problem $\begin{cases}x' = f(t,x) \\ x(0) = 0\end{cases}$ and note $x_0:[0,\omega_0[ \to \mathbb{R}$ its maximal solution.
Show that $\omega_0 = +\infty$ and $\forall t \ge 0. 0 \le x_0(t) \le \gamma(t)$.
How would you prove such a statement, my main tool here (first instant lemma I don't know a standard name for it) does not work here since $x$ is starting at $0$. What guarantee I have that the solution will not fall below $0$?
A diagram describing the situation
