Proof request: Trajectories in the stable manifold of a non-hyperbolic fixed point converge to the fixed point

Question

Let $\dot{x} = f(x)$ be a smooth dynamical system on a finite dimensional vector space $\mathbb{V}$. Suppose $f(0)=0$ is a non-hyperbolic fixed point. For simplicity, I set $Df(0)$ to have a simple $0$ eigenvalue, and all other eigenvalues have negative real part. From center manifold theory, I know there is a unique stable manifold and a possibly non-unique center manifold. Is it true that points starting in the stable manifold are guaranteed to asymptotically converge to $0$? If so, I would love to see a proof of this. I tried searching for one, but am having a difficult time. I would think the proof is similar to the stable manifold theorem, but for sanity's sake would prefer to see a proper proof.