"Show that every closed ball in $\mathbb{R}^n$ is an $n$-dimensional manifold with boundary, as is the complement of every open ball. Assuming the theorem on the invariance of the boundary, show that the manifold boundary of each is equal to its topological boundary as a subset of $\mathbb{R}^n$, namely a sphere. Hint: for the unit ball in Rn, consider the map $\pi \circ \sigma^{-1}: \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\sigma$ 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, I've got (the first part, anyways) of the question done, but my technique was a little different than what Prof. Lee suggested: I considered the ball as infinitely many foliated spheres, mapped each one to a plane using the stereographic projection, and then put the last coordinate as a function of the distance from the north/south pole. However, my solution seemed to neatly avoid any use of $\pi$ as mentioned, and I'm curious if anyone knows how that solution runs.