I am reading Michèle Audin and Mihai Damian's book $\textit{Morse Theory and Floer Homology}$ and I stick at one sentence regarding Global Stable Manifold Theorem which states $W^s(a)$(that is, the stable manifold of $a$) is diffeomorphic to the disk of dimension $n-k$.
In an open subset $U = U(\varepsilon, \eta)$ of $\mathbb{R}^n$ as in Section 2.1.b, we have $W^s(0) = U \cap V_+$ and $W^u(0) = U \cap V_-$. In the notation of Subsection 2.1.b, the stable manifold of $a$ is obtained from the union of $h(U \cap V_+) = h(W^s(0))$ and $h(\partial_+U \cap V+) \times \mathbb{R}$ by identifying $(x, s)$, for $x$ on the boundary and $s \geq 0$, with $\varphi^s(x)$. See Figure 2.3. If $k$ is the index of the critical point $a$, then $h(\partial_+U \cap V_+)$ is a sphere of dimension $n−k−1$; it is the image under the diffeomorphism $h$ of the sphere $\lVert x_+\rVert^2 = \varepsilon$ in $V_+$ (vector space of dimension $n − k$).
Hence the stable manifolds and, likewise, the unstable manifolds, are submanifolds: outside of the critical point, the stable manifold is the image of the embedding $(x, s) \to \varphi^s(x)$ and in the neighborhood of the critical point, it is the image of $V_+$. This argument also shows that $W^s(a)$ is diffeomorphic to the disk of dimension $n − k$
Here $V_+$ stands for the positive subspace of $D^2f(p)$, $x_+$ the coordinate components stand for positive subspace, $U=U(\varepsilon, \eta)$ the bounded domain of local coordinate system (or image of Morse chart), $\partial_+ U = \{x\in U\mid x^TD^2f(p)x=\varepsilon \ \text{and}\ \lVert x_-\rVert^2\leq \eta \}$, and $\varphi^s$ the flow of a pseudo-gradient.
My wonder is: since the above discussion only guarentee diffeomorphisms for region near $p$ and far from $p$ ${\bf separatedly}$, why could we conclude ${\bf a}$ diffeomorphism of $W^s(p)$ to an open disk? Since we have not verified smoothness on the $\partial_+U\cap V_+$.
To tackle this, I tried to construct other diffeomorphisms. ($T^s_pM$ means the positive subspace in my notation)
$\psi$ from $T^s_pM$ to $W^s(p)$: \begin{align} \text{On}\ U\cap T^s_pM\text{:} && \psi(x) &= \varphi_{}(h(\varepsilon\frac{x}{\lVert x\rVert})) &&\\ \text{On}\ V\text{:} && \psi(x) &= \varphi_{\frac{\lVert x\rVert-\varepsilon}{\varepsilon}}(h(\varepsilon\frac{x}{\lVert x\rVert}))&& \end{align} Smoothness on boundary is still a problem.
$\psi$ from $T^s_pM$ to $W^s(p)$: \begin{align} \text{On}\ T^s_pM-\{0\}\text{:} && \psi(x) &= \varphi_{\text{ln}(\frac{\lVert x\rVert}{\varepsilon})}(h(\varepsilon\frac{x}{\lVert x\rVert})) &&\\ \text{At}\ 0\text{:} && \psi(0) &=p && \end{align} Smoothness at $0$ is a problem.
I referred to Augustin Banyaga and David Hurtubise's book $\textit{Lectures on Morse Homology}$ in which the related proof is with very complicated analysis (though the proposition is much stronger, which constructed a diffeomorphism from $T^s_pM$ to $W^s(p)$).
Perhaps I misunderstood Audin and Damian's proof, anyone can help me to figure it out? Thanks in advance.