# Proof of Inverse Function Theorem in Spivak's Calculus on Manifolds

In the proof of Inverse Function Theorem, Spivak states

Lemma 2-10 is given below. Can anyone explain why the statement above is correct? Thanks!

Lemma 2-10 : Let $$A \subset \mathbb{R}^n$$ be a rectangle and let $$f : A \to \mathbb{R}^n$$ be continuously differentiable. If there is a number $$M$$ such that $$| D_j f^i (x) | \leq M$$ for all $$x$$ in the interior of $$A$$, then $$|f(x)-f(y)| \leq n^2 M |x-y|$$ for all $$x,y \in A$$.

Recall right at the beginning of the proof Spivak shows that $$Df(a)$$ is the identity $$\pi$$, so that $$|D_jg^i(x)|=|D_jf^i(x)-D_j\pi^i(x)|=|D_jf^i(x)-D_jf^i(a)|\lt\frac{1}{2n^2}.$$ So, letting $$M=1/2n^2$$, we can apply Lemma 2-10 to conclude $$\big|f(x_1)-x_1-\big(f(x_2)-x_2\big)\big|=|g(x_1)-g(x_2)|\leq\frac{1}{2}|x_1-x_2|.$$
I finally understand it now. The explanation by @Moed Pol Bollo is not accurate. Inequality (3) is correct since $$f$$ is $$C^1$$. Because $$\lambda=Df(a)=I$$, we have $$D_jg^i(x)=D_jf^i(x)-D_jf^i(a)$$. Hence, $$M$$ is actually $$\frac{1}{2n^2}$$ here. Applying Lemma 2-10 yields, $$|g(x_1)-g(x_2)|\le n^2 M|x_1-x_2|=\frac{1}{2}|x_1-x_2|.$$