Serge Lang, in his "Fundamentals of Differential Geometry", states in Lemma 5.4:
Lemma 5.4: Let $U$ be open in $\mathbf{E}$, and let $f: U \rightarrow \mathbf{E}$ be of class $C^1$. Assume that $f(0) = 0$ and $f'(0) = I$. Let $r > 0$ and assume that $\bar{B}_r(0) \subset U$. Let $0 < s < 1$, and assume that $$ |f'(z) - f'(x)| \leq s $$ for all $x, z \in \bar{B}_r(0)$. If $y \in \mathbf{E}$ and $|y| \leq (1 - s)r$, then there exists a unique $x \in \bar{B}_r(0)$ such that $f(x) = y$.
Here, $\mathbf{E}$ is a Banach space, but if you regard it as a finite dimensional normed space, it is fine.
The demonstration sets $g_y(x) = x - f(x) + y$, and uses the Shrinking Lemma to get a fixed point. That is, a point such that $x = g_y(x) = x - f(x) + y$.
The Shrinking Lemma itself, is a bit dynamic: you start at a point and iterates $f$ so it converges to the fixed point. I would like to see this kind dynamic in the context of Lemma 5.4.
So, in order to compare proofs, my question is:
Can you show me a proof that does not use the Shrinking Lemma?
Of course, I do not expect you to simply embed the shrinking lemma inside your proof. Ideally, I would like to see some dynamical construction. Sorry for my subjective requirements. :-)