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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 (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 properness in 1974 and I want to ask about the situation today:

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

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 (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 properness in 1974 and I want to ask about the situation today:

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

Edit:clarification of the question
It is indeed very likely that the answer to the question is "yes", but I would like to be able to quote a complete proof in the literature.
I'm sure that such a proof is technically quite involved since Whitney apparently could not come up with one while Milnor and Stasheff in the quoted book seemed not to be 100% sure that the result is true (cf. their use of the adverb presumably).

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 (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 $\mathbb R^{2n}$ ?

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 (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 properness in 1974 and I want to ask about the situation today:

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

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 $f(X)$ in $\mathbb R^{2n}$.
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 (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 $\mathbb R^{2n}$ ?

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 (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 $\mathbb R^{2n}$ ?