For a smooth manifold $M$ without boundary, it is well known that is admits an exhaustion of compact embedded submanifolds with boundary of codimension zero, i.e. an increasing sequence $(W_n)$ of compact embedded submanifolds with boundary of $M$ of codimension zero such that $M$ is the union of the $W_n$'s.
Does it hold if $M$ has nonempty boundary? Or under what conditions it holds?
As a toy example, if we take $M=X\times[0, 1)$, where $X$ is a smooth compact manifold without boundary, I think taking $M_n=X\times[0, a_n]$, where $\{a_n\}$ is an increasing sequence in $[0, 1)$ converging to 1, would do.