Let $X,Y$ be Banach spaces, $g\in C^1(X,Y)$ and $\overline x\in X$ such that ${\rm D}g(\overline x)$ is surjective.
In Lemma 2.32 of this lecture notes, the following claim is made: There exists $c_g>0$ such that, if $x\in X$ is close enough to $\overline x$, there exists $x'\in X$ such that $$\left\|x'-x\right\|_X\le c_g\left\|g(x)\right\|_Y\text{ and }g(x')=0.\tag1$$
First of all: How exactly is the "if $x$ is close enough to $\overline x$" part influencing $(1)$? Does the author mean that there exists $c_g,\varepsilon>0$ such that $(1)$ holds for all $x\in B_\varepsilon(\overline x)$?
If the statement of the claim is clarified: How can we prove it? Unfortunately the author is only giving a reference in French.