I was going through an example from book Metric Spaces by Searcoid - 3.3.4 (image attached). In this example author assumes condition for 'r' given by r = min {1, |1 - bc|/(1 + |b| + |c|)}. I am not clear, how author come up with this expression. Can anyone help me to figure it out the reason for that assertion?
Note that in this example $\partial \Gamma$ is the boundary of the set $\Gamma$ in $\mathbb{R}^2$
The factor |1 - bc| seems to be the key element here. But I am not sure why that factor was considered to begin with.
