The following is Problem 1-11 in Lee's Introduction to Smooth Manifolds, 2nd Edition:
Let $M = \overline{\mathbb B}^n$, the closed unit ball in $\mathbb R^n$. Show that $M$ is a topological manifold with boundary in which each point in $\mathbb S^{n - 1}$ is a boundary point and each point in $\mathbb B^n$ is an interior point. Show how to give it a smooth structure such that every smooth interior chart is a smooth chart for the standard smooth structure on $\mathbb B^n$.[Hint: consider the map $\pi \circ \sigma^{-1}: \mathbb R^n \to \mathbb R^n$, where $\sigma : \mathbb S^n \to \mathbb R^n$ is the stereographic projection and $\pi$ is a projection from $\mathbb R^{n + 1}$ to $\mathbb R^n$ that omits some coordinate other than the last.]
So, for the first part of the question I proceeded as follows:
It is clear that for each point in $\mathbb B^n$ there is a ball $B$ of small radius such that $(B, Id + (0, \ldots, 0, 1))$ is a coordinate chart onto $\text{Int} \mathbb H^n$. So each point in $\mathbb B^n$ is an interior point.
Now, how to show the other two claims? How to use the hint?
Thanks in advance and kind regards.
