[proofplan]
The proof has three stages. Cartan's Theorem A produces enough global holomorphic functions to separate points and tangent vectors, giving a holomorphic immersion that is injective on chosen compact exhaustion levels. A rapidly growing sequence of additional global functions makes the map proper. A generic linear projection then lowers the target dimension to $2n+1$ while preserving properness and injectivity.
[/proofplan]
[step:Construct global functions separating points and tangent vectors]
Let $X$ be a Stein manifold of dimension $n$. For distinct points $p,q\in X$, Cartan's Theorem A applied to the ideal sheaf of $p$ and the quotient recording the value at $q$ gives a global holomorphic function separating $p$ and $q$. Applying Theorem A to the square of the maximal ideal at a point gives global functions whose first-order germs generate the cotangent space. Hence one can choose countably many global holomorphic functions whose differentials span $T_p^*X$ at every point and whose values separate points.
[/step]
[step:Obtain an injective holomorphic immersion into a large Euclidean space]
Choose a Stein exhaustion $K_1\subset K_2\subset\cdots$ of $X$. On each compact $K_j$, finitely many of the separating functions suffice to separate all pairs of points in $K_j$ and to span cotangent spaces on $K_j$. By adding functions inductively, one obtains a holomorphic map
\begin{align*}
F:X\to\mathbb{C}^N
\end{align*}
for some finite or countable $N$ that is injective and has injective differential on each exhaustion level. Standard diagonal selection gives a finite-dimensional map after enlarging $N$ enough on successive exhaustion steps.
[/step]
[step:Force properness using the Stein exhaustion]
Let $\rho:X\to\mathbb{R}$ be a strictly plurisubharmonic exhaustion. Cartan's Theorem A and the Oka-Weil approximation theorem allow one to construct global holomorphic functions $g_j$ that are small on $K_j$ and large on $X\setminus K_{j+1}$. Adding these functions as coordinates makes the resulting map $G:X\to\mathbb{C}^M$ proper: if a set in $\mathbb{C}^M$ is bounded, the growth of the $g_j$ forces its inverse image to lie in some compact $K_j$.
[/step]
[step:Project to $\mathbb{C}^{2n+1}$]
The map $G$ is a proper injective immersion into some high-dimensional affine space. For a generic complex linear projection $\pi:\mathbb{C}^M\to\mathbb{C}^{2n+1}$, the restriction $\pi\circ G$ remains an immersion and remains injective. The reason is dimension counting: the tangent variety has dimension at most $2n$, and the secant variety of pairs of distinct points has dimension at most $2n+1$; a generic kernel avoids the bad directions. Properness is preserved under a projection chosen injective on the relevant closed image. Thus $\pi\circ G:X\to\mathbb{C}^{2n+1}$ is a proper holomorphic embedding.
[/step]
