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A fixed point $x_0$ is Lyapunov stable if for every $\epsilon >0$ there is $\delta >0$ such that solutions starting within $\delta$ of $x_0$ stay within $\epsilon$ for all forward times. It has quasi-asymptotic stability if some neighborhood of $x_0$ consists of initial points whose solutions converge to $x_0$. Asymptotic stability requires both properties.

Choose the Lyapunov functionIt vanishes only at the origin, is positive elsewhere, and has compact sublevel sets. The vector field is polynomial, hence locally Lipschitz. Along a solution,A small sublevel set is forward invariant and can be fitted inside any prescribed neighborhood, proving Lyapunov stability; boundedness also prevents finite-time escape.

For attraction, a bounded forward solution has a nonempty compact limit set, and $V$ decreases to a limit. Every point in that limit set has that same $V$ value. Continuous dependence of solutions makes the limit set forward invariant, so $\dot{V}=0$ along every trajectory in it. Such a trajectory must have $x=0$ at all times. But then $\dot{x}=-y$, forcing $y=0$. Thus the only possible limit point is the origin, and every solution converges there. This verifies the invariant-set condition behind the LaSalle invariance principle, rather than incorrectly treating $\dot{V}$ as strictly negative at every nonzero point. The origin is therefore asymptotically stable, indeed globally attracting.