I have the following problem:
Let $S^n=\{x\in\mathbb{R}^{n+1}:\|x\|=1\}$ and $\pi_{\pm}:S^n\setminus\{{p_{\pm}}\}\to\mathbb{R}^n$ be the stereographic projections from the poles $p_{\pm}=(0,...,\pm 1)$. Then $$\mathcal{A}=\{(S^n\setminus\{{p_+}\},\pi_+),(S^n\setminus\{{p_-}\},\pi_-)\}$$
is a maximal atlas in class $C^{\infty}$.
I already showed that $\pi_+$ and $\pi_-$ are homeomorphism. Obviously the sets $S^n\setminus\{{p_{\pm}}\}$ cover $S^n$. And $\pi_+$ and $\pi_-$ are $C^{\infty}$-compatible.
The hard thing is that I really don't know how to show this atlas is maximal. I mean, if we take $(U,\phi)$ any chart $C^{\infty}$-compatible with both $(S^n\setminus\{{p_+}\},\pi_+),(S^n\setminus\{{p_-}\},\pi_-)$, how to show $(U,\phi)\in\mathcal{A}$?
Thank you.