Help to understand the proof about injective immersions

Question

I'm currently working on the proof of the following theorem in the book of "Bröcker":

Let $M^m$ be a differential manifold and $f:M^m \to \Bbb R^n$ be a differentiable map and $2m < n$. Let $A \subset M$ be closed and let the restriction of $f$ to a neighbourhood $U$ of $A$ be an injective immersion. Then arbitrarily close of $f$ there exists an injective immersion $g: M^m \to \Bbb R^n$, s.t. $g \vert A = f \vert A$.

The proof is given as follow:

I don't understand here why, we have that $g_v$ are still immersions?

Also later, he used the Sard theorem to implies that the image of this map has measure zero. So we may choose $b_v$ not to be in this image.. I don't understand how he used here the theorem and how he can conclude that.

EDIT:

Here is the Lemma 7.6 but I don't see the relation.

Many thanks for some help.

Answer 1

First, if $g_\nu$ has rank $m=\dim M$, then that means the map is an immersion.

Second, a set of measure $0$ has empty interior, and so the complement is dense. It certain follows that the complement is non-empty.