## embedding of a compact manifold into R^n?

I have problems to come up with a proof of this, but i am pretty sure that it is correct: Let $M$ be a compact n-manifold with boundary the disjoint union $\partial M=M_1 \cup M_2$. By Whitney's embedding theorem we can embed the boundary into $\mathbb{R}^{2n}$. Is it true that we can extend this to an embedding of $M$ into $\mathbb{R}^{2n} \times [0,1]$? And secondly in a way such that $M \cap \mathbb{R}^{2n} \times 0=M_1$ and $M \cap \mathbb{R}^{2n} \times 1=M_2$.

I think the first claim should proof well, but i am not sure about the second. Grateful for any answers.

banach