This question concerns problem 1-B in the book of Milnor and Stasheff part a. They first define the set $F:=\{f:\mathbb{R}P^n\rightarrow\mathbb{R} \mid \text{$f\circ q$ is smooth}\}$ where $q:\mathbb{R}^{n+1}-{0}\rightarrow\mathbb{R}P^n$ that sends $x$ to $\mathbb{R}x$. The problem is to show that $F$ is a smoothness structure on $\mathbb{R}P^n$.
It is my understanding that to do this I must do the following: First I should show that the set of functions in $F$ separates points on $\mathbb{R}P^n$, then show that $i(\mathbb{R}P^n)\subset\mathbb{R}^{F}$ is a smooth manifold where $i_{f}(x)=f(x)$ for $f\in F$. Finally I should show that $F$ is the set of all smooth real valued functions on $\mathbb{R}P^n$.
1) What are some candidates for functions in $F$ that separate points? My first guess was maybe I should play with trignometric functions but then I think smoothness becomes an issue. Perhaps the $f_{ij}$ functions defined in part b work?
2) I dont know how to get my hands on $i(\mathbb{R}P^n)$ because $F$ could be infinite. What charts can you use? (I know what charts you should use if you use the definition of manifold given in a book like Lee's).
I hope things are clear, thanks!