Prove $x\to 0$ as $t\to \infty$ if we consider the system of equations $x'=(A+B(t))x$ where $B(t)\to 0$ and $A$ has negative eigenvalues.

Question

Consider a matrix $A$ such that all of its eigenvalues are negative. Consider $B(t)$ where $B(t)\to 0$ as $t\to\infty$. Then consider the system of equations

$$ x'=(A+B(t))x$$ Prove that any solution $\phi$ to this system must have the property $\phi(t)\to 0$ as $t\to\infty$. If we ignore $B$ I see that the negative eigenvalues of $A$ cause the solutions to converge to $0$, but how do you solve this for a general $B(t)$?