Large-scale Lipschitz and bornologous maps

Question

I'm trying to understand the following questions which I took from Roe's book "Lectures on coarse geometry".

A map of metric spaces $f:X \to Y$ is called large-scale Lipschitz if there are positive constants $A$, $c$ such that

$\rho(f(x), f(y)) \leq c\rho(x, y) + A$

A map $f$ is named bornologous if for every $R>0$ there is $S>0$ such that

$\rho(x,y)<R \Rightarrow \rho(f(x),f(y))<S$

My questions:

1. Why if the map $f$ is large-scale Lipschitz then this map is bornologous?
2. Example where the converse is not true. When bornologous map $f$ is not large-scale Lipschitz map for a map of metric spaces $f:X \to Y$
3. As I understand that bornologous map and large-scale Lipschitz map become equivalent definitions in geodesic metric spaces. Why is it true?

Can you explain, please, this question in more details?

Thank you!

Answer 1

Regarding question 1, assume that $f$ is large scale Lipschitz with constants $A,c$. To prove $f$ is bornologous, given $R>0$ let $S=cR+A$. It follows that if $\rho(x,y)<R$ then $$\rho(f(x),f(y)) \le c \rho(x,y) + A < cR+A = S $$

Regarding question 2, take $X=\mathbb N$ and $Y = \{2^n \mid n \in \mathbb N\}$ and let $f(n)=2^n$.

Regarding question 3, assuming that $f$ is bornologous, let $R=1$, choose a corresponding $S$, and then let $c=A=S$.