There are a few bits of evidence that will give you the proof if you put them together in the right way.
Can you show that $x(t)\equiv 1$ is a solution of the ODE?
Can you show that $x(t)\equiv 0$ is a solution of the ODE?
What does the uniqueness part of this theorem tell you about a solution (considered as a curve in the $t-x$ plane) that passes through a point of the form $(t_0,1)$? Same question for $(t_0,0)$?
Can a solution passing through the point $(t,x)=(0,x_0)$, with $x_0\in(0,1)$, ever meet a point of the form $(t_0,1)$ for some $t_0\in\mathbb{R}$? ...of the form $(t_0,0)$?
What is the sign of $\frac{dx}{dt}$ for a solution $x(t)$?
What can you say about a function $t\to x(t)$ that is increasing and bounded above by 1?
Putting everything together you should be able to get this: the solution $u$ of the problem is increasing and bounded above by 1, and therefore has a limit $\ell\leq 1$ satisfying $$\lim_{t\to\infty}u(t)=\ell.$$
Now consider what happens if $\ell<1$ - in particular, what would this tell you about $f(\ell)$?