Let $M$ be a smooth, n-dimensional manifold. Prove that for every $k \leq n$ there exists an embedding $ \mathbb{R}^k \to M$.
I'm having trouble visualising this. How can $\mathbb{R}^2$ be embedded into the 2-dimensional torus, considering that the torus is a compact manifold and $\mathbb{R}^2$ is certainly not. $f(\mathbb{R}^2)$ is supposed to be a submanifold of the torus and there are not that many options left. Is there something obvious I'm forgetting?
An open disc on any surface is always diffeomorphic to the whole $\Bbb{R}^2$. To see that, take for example the smooth, invertible map (in polar coordinates): $$ \theta \mapsto \theta\\ r\mapsto \arctan r. $$