Proof of the weak Whitney Embedding theorem

Question

I was studying 'Introduction to smooth manifolds' by John M.Lee(GTM218) when I encountered this problem.
A weak version of Whitney Embedding theorem is stated as follows:
Every smooth n-manifold $M$ with or without boundary admits a proper smooth embedding into $R^{2n+1}$, in which smooth embedding means a topological embedding with the property that at each point its differential is injective.
In the book, the proof is as follows:
First we treat the case when $M$ is compact. One finds a finite covering by regular coordinate balls, use them to create an embedding into an Euclidean space, then using the Sard's theorem one gradually reduce its dimension until it reach $2n+1$. Then we treat the noncompact case. We find a exhaustion function $f$(a smooth function $f$ from $M$ into $R$ with the property that for every real number $c$ the set $f^{-1}(-\infty,c]$ is compact), use the Sard's theorem to find (for every nonnegative integer $i$) regular values $a_{i}$ and $b_{i}$ such that $$i\lt a_{i}\lt b_{i}\lt i+1$$ define $$D_{0}=f^{-1}(-\infty,1],E_{0}=f^{-1}(-\infty,a_{1}],D_{i}=f^{-1}[i,i+1],E_{i}=f^{-1}[b_{i-1},a_{i+1}]$$ Notice from Proposition 5.47 that each $E_{i}$ is a compact regular domain, let $\rho_{i}:M\rightarrow R$ be a smooth bump function that is equal to $1$ on a neighborhood of $D_{i}$ and supported in $Int E_{i}$, define $$F:M\rightarrow R^{2n+1}\times R^{2n+1}\times R$$ by $$F(p)=(\sum_{i\ even}\rho_{i}(p)\phi_{i}(p),\sum_{i\ odd}\rho_{i}(p)\phi_{i}(p),f(p))$$ One easily verifies that $F$ is a proper smooth embedding, so we could reduce the dimension of the codomain from $4n+3$ to $2n+1$, which concludes the proof.
The problem is, Proposition 5.47 states that if $M$ is a smooth manifold $\textbf{without boundary}$, $f$ a smooth function from $M$ to $R$, and $a\lt b$ regular values of $M$,than $f^{-1}[a,b]$ is a regular domain. In fact, if we replace 'without boundary' by 'with boundary', then the proposition is false, as is easily seen by the counterexample $M=H^{2}=\{{(x,y)|y\geq 0}\},f(x,y)=x,b=0$. So the proof breaks here.
Does anyone have any idea on how to fix this problem? Any hints will be helpful.