Show $\varphi(x_1,...,x_{n+1})=\frac{1}{1-x_{n+1}}(x_1,...,x_n)$ is surjective.

Question

We consider in fact the stereographic projection. Let $$\mathbb S^{n}=\{x_1^2+...+x_{n+1}^2=1\mid (x_1,...,x_{n+1})\in\mathbb R^{n+1}\}$$ and $E=\text{span}(e_{n+1})^\perp$ where $e_{n+1}=(0,...,0,1)$. Let $$\varphi:\mathbb S^{n+1}\backslash \{e_{n+1}\}\longrightarrow E$$ defined by $$\varphi(x_1,...,x_{n+1})=\frac{1}{1-x_{n+1}}(x_1,...,x_n).$$ I have to show that $$\big\{(\mathbb S^{n+1}\backslash \{e_{n+1}\},\varphi)\big\}$$ is an atlas. In the solution, it's written that the fact that $\varphi$ is surjective is obvious. Is it really that obvious ? I mean, if $\varphi:\mathbb R^{n+1}\backslash \{e_{n+1}\}\longrightarrow E$, I would say that the surjectivity is obvious... but here, it doesn't look that obvious (in any case, not for me).

Answer 1

If it doesn't seem obvious, you can pick a point $x$ in $E$ and construct its preimage. Hint: parametrize the line that goes through $x$ and $e_{n+1}$. It's also useful to think about it in the low dimensional case, and then generalize.