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.