How to show $\mathbb{R}^{n}+\mathbb{S}^{n}=\mathbb{R}^{n}$?

Question

I want to ask how to show $\mathbb{R}^{n}+\mathbb{S}^{n}\cong \mathbb{R}^{n}$ as connected sums where the isomorphism is a differeomorphism between $\mathbb{R}^{n}$ and $\mathbb{R}^{n}$. The proof in Kosinski's book is not readable. Sorry if this problem is too trivial.

Answer 1

The addition happens like this: You remove a disc from $\Bbb R^n$, and one from $\Bbb S^n$, and glue the two of them together along the edge of the hole you just opened. But if you remove a disc from $\Bbb S^n$, what's left is diffeomorphic to a disc, so what you are really doing to your $\Bbb R^n$ is removing a disc, and then glueing a new disc onto the hole. All in all, you're back to $\Bbb R^n$