A generic projection can preserve properness

The injectivity and immersion assertions follow exactly as in the generic-projection lemma recorded in [F1]: since $u$ is not a secant direction, distinct points cannot collapse under $P_u$, and since $u$ is not a tangent direction, no nonzero tangent vector lies in the kernel of $d(P_u\circ F)$.

Let $e=(0,1)$ be the last-coordinate unit vector and put $e^{\mathrm{\prime }}:=P_u(e)$. The hypothesis that $u$ is not parallel to $e$ is exactly $e^{\mathrm{\prime }}\mathrm{\ne }0$. Decompose $u^{\perp }=\mathbb{R}{e}^{\mathrm{\prime }}\oplus (e^{\mathrm{\prime }}{)}^{\perp }$. Since $P_u(F(p))=P_u(g(p),0)+\rho (p)e^{\mathrm{\prime }},$ its $(e^{\mathrm{\prime }}{)}^{\perp }$-component is bounded. Its scalar component along $e^{\mathrm{\prime }}\mathrm{/}\mathrm{\parallel }{e}^{\mathrm{\prime }}\mathrm{\parallel }$ is ${\rho }^{\mathrm{\prime }}(p)=\mathrm{\parallel }{e}^{\mathrm{\prime }}\mathrm{\parallel }\rho (p)+b(p),$ where $b$ is bounded.

The function ${\rho }^{\mathrm{\prime }}$ is proper. Indeed, if $J\subseteq \mathbb{R}$ is compact and $\mathrm{\mid }b\mathrm{\mid }\le B$, then ${\rho }^{\mathrm{\prime }}(p)\in J$ forces $\rho (p)$ into a bounded closed interval because $\mathrm{\parallel }{e}^{\mathrm{\prime }}\mathrm{\parallel }>0$. Thus $({\rho }^{\mathrm{\prime }}{)}^{-1}(J)$ is a closed subset of the inverse image under the proper map $\rho$ of a compact interval.

If $K\subseteq u^{\perp }$ is compact, its image under the linear coordinate along $e^{\mathrm{\prime }}$ is compact. Hence $(P_u\circ F{)}^{-1}(K)$ is a closed subset of the compact set $({\rho }^{\mathrm{\prime }}{)}^{-1}({\mathrm{pr}}_{e^{\mathrm{\prime }}}K)$ and is compact. Therefore $P_u\circ F$ is proper. Together with step 1.1, it is a proper injective immersion.