= First Lyapunov theorem
{c}
A positive-definite <Lyapunov function> with nonpositive orbital derivative proves <Lyapunov stability> of the equilibrium. Indeed, fix a sufficiently small ball $B_\varepsilon$ in the function's domain and put
$$
\alpha=\min_{\lVert x\rVert=\varepsilon}V(x)>0.
$$
Continuity and $V(0)=0$ give a ball $B_\delta$ on which $V<\alpha$. Since $V$ cannot increase along a trajectory, one starting in $B_\delta$ cannot cross the sphere $\lVert x\rVert=\varepsilon$, on which $V\geq\alpha$. It therefore remains in $B_\varepsilon$ for all forward time.
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