Ok, So I am reading a theorem in Lee's ''Introduction to smooth manifolds''.
The lemma states as follows: ''Let $F:M \to \mathbb{R}^n$ be any smooth map, where $M$ is an m-manifold and $n\geq 2m$. For any $\epsilon > 0$, ther is a smooth immersion $\tilde{F}: M \to \mathbb{R}^n$ such that $sup_m \mid \mid \tilde{F}-F \mid \mid \leq \epsilon$.''
Now I really do not understand to completion how the proof goes, so there may be something I am missing in this:
As far as I can tell, the hypothesis of $n \geq 2m$ is only used to find a $n \times m$ matrix $A$ such that $\mid \mid A \mid \mid < \delta$ for a given $\delta > 0$, and such that $A$ is not of the form $A=B - d_xF$ with $B$ of rank less than $n$. (There are some subtleties which I am skipping, like where $x$ is at, but are not important for what I am propposing).
Which can be done by finding an injective matrix $B$ close enough to $d_xF$.
So, if space of real matrices of rank $m$, dense in $M_{n \times m}(\mathbb{R})$ , then you can always find an injective matrix $B$, such that $\mid \mid B-d_xF \mid \mid < \delta$, for any given $\delta$, which would mean that this lemma is true for $n > m$. So, I do not know if its the proof I am not understanding or this statement is false (Which I proved, but there can be something wrong in my proof.)