A celebrated theorem of Nash is that every $C^k$ ($k\geq 3$) Riemannian manifold $(M,g)$ can be isometrically embedded into some Euclidean space $\mathbb{R}^d$ for some $d\in \mathbb{N}$. However, I have not been able to track down any results when $M$ is a manifold with boundary. Is there a result of the same ilk when $M$ has nonempty boundary?

Perhaps more specifically, is it true that any Riemannian manifold with boundary can be isometrically and properly embedded into some $\mathbb{R}^d$? (Properly is included so that in particular, the image of the boundary corresponds to the topological closure of the embedded manifold in $\mathbb{R}^d$.)

This seems like a natural question to ask, and I was wondering if it has already been studied.