= Solution
Let $x_*$ be an <equilibrium point of a dynamical system>. A <Lyapunov function> on a neighbourhood of $x_*$ is a continuously differentiable function $V$ such that
$$
V(x_*)=0,\qquad V(x)>0\quad(x\ne x_*),
$$
and whose derivative along trajectories satisfies
$$
\dot V(x)=\nabla V(x)\cdot f(x)\leq0.
$$
The <First Lyapunov theorem> says that the existence of such a positive-definite $V$ proves <Lyapunov stability> of $x_*$. If $\dot V<0$ away from $x_*$, the <Second Lyapunov theorem> strengthens this to <asymptotic stability>.
<LaSalle invariance principle> says that if a trajectory remains in a compact positively invariant set $\Omega$ on which $\dot V\leq0$, then it approaches the largest invariant subset of
$$
\{x\in\Omega:\dot V(x)=0\}.
$$
In particular, if that largest invariant subset consists only of $x_*$, every trajectory in $\Omega$ approaches $x_*$.
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