Let $\displaystyle M$ be a non-empty differentiable manifold of dimension $\displaystyle n$. Prove there exists an embedding $\displaystyle f: \mathbb{R} ^k \longrightarrow M$ for all $\displaystyle 1 \leq k \leq n$.

For the sphere $\displaystyle \mathbb{S} ^n$ it's clear that the stereographic projection is an embedding, but I can't prove the general case. Any ideas would be apreciated.