To prove that equilibrium point of a dynamical system is stable

Question

I am trying to prove that the fixed point of following first order ODE is stable.

$\dot x(t)=a(t)-b(t)x(t),\quad x(0)=x_0>0$

We know that $x(t)$ is nonnegative and uniformly bounded above a priori.

Also, we have the following property (*)

(*): $a(t),b(t) \in A \subset (0,\infty),~\forall t$ for some bounded interval $A$.

I want to prove that $\lim_{t \to \infty}\dot x(t)=0$.

I tried to show that $\lim_{t \to \infty}x(t)$ exists and apply Barbalat's lemma.

I want to get rid of property (*) but i'm assuming it at this stage. Thank you in advance.