The weak Whitney proper embedding theorem

Choose the embedding $F:M\to {\mathbb{R}}^d$ from [L1]. If $M$ is compact, $F$ is automatically proper. If $M$ is noncompact, use the supplied form $F=(G,\rho )$ relative to a decomposition ${\mathbb{R}}^d=e^{\perp }\oplus \mathbb{R}e$, with $G$ bounded and $\rho$ proper.

If $d\le 2n+1$, compose $F$ with a linear isometric inclusion ${\mathbb{R}}^d\hookrightarrow {\mathbb{R}}^{2n+1}$. This composite is still a proper smooth embedding, so the theorem is proved in this case. Hence assume $d>2n+1$.

The generic-projection lemma in [L2] gives a dense set of directions $u\in S^{d-1}$ for which $P_u\circ F$ is an injective immersion. If $M$ is compact, choose any such $u$; the projected map is proper because its source is compact. If $M$ is noncompact, the set $S^{d-1}\setminus \{e,-e\}$ is a nonempty open set, so it meets the dense good-direction set. Choose $u$ in that intersection. Then $u$ is not parallel to the proper-coordinate axis, and the corrected properness lemma in [L2] makes $P_u\circ F$ proper. In either case [L3] upgrades the proper injective immersion to a smooth embedding into $u^{\perp }\cong {\mathbb{R}}^{d-1}$.

In the noncompact case put $e^{\mathrm{\prime }}:=P_u(e)\mathrm{\ne }0$ and decompose $u^{\perp }=\mathbb{R}{e}^{\mathrm{\prime }}\oplus (e^{\mathrm{\prime }}{)}^{\perp }$. The component of $P_u\circ F=P_u(G)+\rho e^{\mathrm{\prime }}$ perpendicular to $e^{\mathrm{\prime }}$ is bounded. Its scalar component along $e^{\mathrm{\prime }}\mathrm{/}\mathrm{\parallel }{e}^{\mathrm{\prime }}\mathrm{\parallel }$ is $\mathrm{\parallel }{e}^{\mathrm{\prime }}\mathrm{\parallel }\rho +b$ with $b$ bounded, and the argument in the properness lemma shows this function is proper. Thus the projected embedding again has bounded-plus-proper form, now with proper-coordinate unit vector $e^{\mathrm{\prime }}\mathrm{/}\mathrm{\parallel }{e}^{\mathrm{\prime }}\mathrm{\parallel }$.

If the new ambient dimension is still greater than $2n+1$, repeat steps 3.1-4.1. Step 3.1 restores the embedding hypothesis after each projection; compactness preserves properness in the compact case, and step 4.1 preserves the bounded-plus-proper form in the noncompact case. After the finite number $d-(2n+1)$ of projections, the ambient dimension is $2n+1$ and the resulting map is a proper smooth embedding.

The low-dimensional branch is step 2.1, and the projection branch is step 5.1. Therefore every smooth $n$-manifold admits a proper smooth embedding into ${\mathbb{R}}^{2n+1}$.