I'm reading the proof of inverse theorem from Hubbard's vector calculus book. In the proof, the authors use a lemma (Lemma A7.1).
Let B be the open unit ball of $R^n$, and let $F: B \rightarrow R^n$ be a $C^1$ mapping such that $F(0) = 0$, $[DF(0)]= I$, and $|[DF(x)] - [DF(y)]| \leq |x -y|$ for all $x, y \in B$. Then F is injective.
The following is the proof included in the book.
\begin{align} |F(x) - F(y)| &= |x - y + (F(x) - x) - (F(y) -y)|\\ & = |x - y + (F - I)(x) - (F - I)(y)|\\ & \geq |x - y| - sup_{z\in[x,y]}|[DF(z)] - I||x-y|\\ & = |x - y| - sup_{z\in[x,y]}|[DF(z)] - [DF(0)]||x-y|\\ & \geq |x - y| - sup_{z\in[x,y]}|z-0||x-y|\\ &=|x-y|(1 - sup(|x|, |y|))\end{align}
Thus $F(X) = F(Y)$ implies $x =y$, or F is injective.
Line 3 is due to a corollary of mean value theorem: $|F(b) - F(a)| \leq sup_{c \in [b,a]}|[DF(c)]||\vec{b-a}| $.
Line 4 is due to the assumption $[DF(0)]= I$.
Line 5 is due to the assumption $|[DF(x)] - [DF(y)]| \leq |x -y|$ for all $x, y \in B$.
Line 6 is due to $sup_{z \in [x,y]}(|z|) = sup(|x|,|y|)$ since the point of a line segment farthest from the origin is always one of the endpoints.
I just couldn't understand where the fact $F(0) = 0$ is used.!