TITLE: Why is the $n$-sphere a Smooth Manifold?
QUESTION [5 upvotes]: I want to know how the $n$-sphere is a smooth manifold? 
I'm unable to understand the theory of differentiable structure on the $n$-sphere. 
Please tell me any suggested reading for a good start on differentiable manifolds. 
Presently I am following John M. Lee's book which is quite difficult for me to understand.

REPLY [5 votes]: If you're looking for a more explicit description of an atlas (rather than a more implicit proof that it is a smooth manifold, like using the submanifold theorem in TooOldforMath's answer, although that is certainly the easier way of proving it), the standard way is to choose the following cover:
$$ U_1=S^n\setminus \{(0,0,\cdots,0,1\} \text{ and } U_2=S^n\setminus \{(0,0,\cdots,0,-1\}, $$
and use stereographic projection as the homeomorphisms from $U_1$ and $U_2$ to $\mathbb{R}^n$.  This explicitly defines an atlas.