Trying to prove that $\displaystyle S^n=[x\in{R^{n+1}}:||x||=r]$ is a manifold.

I think I can see why it's Hausdorff, take two points on the sphere of distance $\displaystyle \epsilon$ from each other, then 'draw' two open balls around both these points of radius less than $\displaystyle \frac{\epsilon}{2}$, so it should be Hausdorff.

I'm not sure about proving 2nd countability, draw open balls of radius 1 around all the points with rational coordinates?

And I've no idea about proving it's locally Euclidean.

And finally, just to clarify I'm not doing this wrong, then open sets of $\displaystyle S^n$ are the same open sets used in $\displaystyle R^n$? (i.e. the open sets as given by the distance metric?)