I'm studying Matsumoto's An Introduction to Morse Theory. I want to solve a problem on page 76.
Context: Let $M$ be a closed $m-$manifold and $f:M\rightarrow \mathbb{R}$ a Morse function. Let $c$ be a critical value of index $\lambda$ and $p\in M$ such that $f(p)=c$. By Morse's lemma, there exists a neighborhood about $p$ where $f$ has the standard form:
$$f=-x_1^2-\cdots -x_{\lambda}^2+x_{\lambda +1}^2+\cdots + x_m^2+c.$$
Consider the set $H$ (a $\lambda$ -handle) determined by the inequalities
$$x_1^2+\cdots +x_{\lambda}^2-x_{\lambda +1}^2-\cdots - x_m^2 \leq \epsilon ~,~ x_{\lambda +1}^2+\cdots + x_m^2\leq \delta ,$$
where $\epsilon$ is sufficiently small and $\delta$ is much smaller than $\epsilon$.
Problem: Prove that $H$ is diffeomorphic to $D^{\lambda}\times D^{m-\lambda}$.
I don't know how I can prove this. In particular, I know there are problems with the corners. I thought of using the following theorem:
Theorem: Let $M$ be a smooth manifold with corners. Then there exists a smooth manifold $M_0$ without corners that is homeomorphic to $M$ and diffeomorphic to $M$ outside of a neighborhood of the corner points. Furthermore, $M_0$ is unique up to diffeomorphism.
Any ideas?