I am trying to show this using the definition:
M is a k-dimsensional submanifold of $\mathbb{R^n}$ if for all $x \in M$ the following condition holds:
There exists an open set $U \subset \mathbb{R^n}$ with $x \in U$, and an open set $V \subset \mathbb{R^n}$ with a diffeomorphism $h: U \to V$ such that $$h(U \cap M) = V \cap (\mathbb{R^k} \times \{0\}^{n-k}) = \{ y \in V: y^{k+1} = \dots = y^n = 0\}$$
I've said, that since $M$ is open, we can draw a ball around every $x \in M$, and I wanted to let $U = \cup_x B_x$ which is open, but I am not sure how to construct $V$ or $h$.