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

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

banach