Whitney embedding theorem

Statement
The Whitney embedding theorem states that any compact connected differential manifold of dimension $$n$$ possesses a smooth embedding into $$\R^{2n+1}$$. By smooth embedding, we mean it can be viewed as a subspace, with the subspace topology, and further, that the induced mapping of tangent spaces is also injective.

Proof ingredients
Two ingredients are used in the proof:


 * Compactness allows us to work with a finite atlas, and consider a partition of unity

We can use Sard's theorem to predict certain properties of maps that we construct.
 * Sard's theorem, or rather, the following corollary of Sard's theorem: if $$m < n$$, the image of any $$m$$-dimensional manifold in a $$n$$-dimensional manifold via a differentiable map, has measure zero in the latter.