Differential equations - bounded dynamical system

Question

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be $C^1$-function and let $I_{x_0}=(a,b)$. Assume that there exist $M>0$ such that $|\varphi(\cdot,x_0)|_{[0,b)}|\le M$, where $\varphi(t,x)$ is dynamical system generated by equation $x'=f(x)$. How to show that $b=+\infty$ and there exist $\lim_{t\to\infty}\varphi(t,x_0)$ ?

Answer 1

Hint: Any solution to $\dot x=f(x)$ is a monotone function.