The Stable Manifold Theorem Applications Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$
Where $S$ is a $k$-dimensional differentiable manifold tangent to the stable subspace $E^s$ of the linear system $x'=Ax$ where $A=Df(0)$ and $0$ is a hyperbolic equilibrium point.
Show that $W^s(0)$ is unique and invariant with respect to the flow $\phi_t(x)$ ; furthemore, for all $x\in W^s(0)$, $$\lim_{t \rightarrow \infty} \phi_t(x)=0.$$

Both properties are immediate from the definitions. Note that $S$ is unique is a sufficiently small neighborhood and so $W^s(0)$ is well defined, that is, the set $W^s(0)$ is independent of $S$. It is also invariant since $\phi_t(S)\subset S$ for all positive $t$.
The last property follows from the definition: indeed, for each $x\in W^s(0)$ there exists $t>0$ such that $\phi_t(x)\in S$ (by the definition of $S$) and so $\phi_s(x)=\phi_{s-t}(\phi_t(x))\to 0$ when $s\to+\infty$ (since this happens if and only if $t-s\to+\infty$.
