If $X$ is a $ C^\infty$ differential manifold of dimension $n$, then there exists an embedding $f:X\to \mathbb R^{2n+1}$.   
This is a not too difficult  theorem due to Whitney, proved in many textbooks.  
The interesting point is that $f$ may be chosen to be a proper map: the inverse image of a compact subset  in $\mathbb R^{2n+1}$ is compact in $X$ or, equivalently, $f$ has closed image $f(X)\subset \mathbb R^{2n+1}$.   
Now, many books go on to add that actually Whitney proved that $X$ may be embedded into $\mathbb R^{2n}$, but that the proof is much more difficult. (I don't know any book with a  proof of this sharper result)   

I had always assumed that in the  sharper result the embedding was also proper but I was astonished to read, a few days ago, in Milnor-Stasheff's [Characteristic Classes](http://books.google.fr/books/about/Characteristic_Classes.html?id=5zQ9AFk1i4EC&redir_esc=y) (page 120) that in the embedding into  $\mathbb R^{2n}$ one may  **"presumably"**  assume that the image of $X$ closed, but that it is not proved in Whitney.  
So  Milnor (one of the greatest topologists of all times!) didn't seem to be sure about propernes in 1974 and I want to ask about the situation  today:

**Question**   
Can an $n$-dimensional differential manifold be **properly** embeded into $\mathbb R^{2n}$ ?