Why is this function Lipschitz? Let $f:A \to B$ where $A$, $B \subset \mathbb{R}^n$. 
Suppose
$$\lVert f(y_1) - f(y_2)\rVert_{\ell_\infty} \geq C\lVert y_1 - y_2 \rVert_{\ell_\infty}$$
This tells us that $f$ is one to one and that the inverse is Lipschitz.
I am told that $f$ is bi-Lipschitz; so $f$ is also Lipschitz, but I don't see why?

I was inaccurate: in the general settings of your problem, where the only thing we know on $A$ is that $A \subset \mathbb{R}^n$, it is not true that even if $f \in W^{1,\infty}(A)$ then $f \in \text{Lip}(A)$. Neverthless what is true is the following:
$\textbf{Theorem:}$ Leu $U$ be open and bounded, with $\partial U$ of class $C^1$. Then $u \colon U \to \mathbb{R}$ is Lipschitz continuous if and aonly if $u \in W^{1,\infty}(U)$.
This is proved in Evan's Partial Differential Equations: this is Th. 4 of the additional topics in chapter 5.
For the case in which $U$ is unbounded you just need to read the section dedicated to extentions.
The theoreom is pretty good, indeed any weakening of the hypothesis makes the theorem false: google has counterexamples. (Even the Hypothesis $\partial U$ of Lipschitz class is too weak.)
If your $A$ is a general open set what is true is that $u \in W^{1,\infty}_{loc}(A) \Longleftrightarrow u \in \text{Lip}_{loc}(A)$.
I really hope this helps!
