Show that the distance ρ, between two points with the same ordinate on the lines x = 0 and x = 1, goes to 0 when y approaches infinity.

Question

I want to show that in the half-plane model the distance ρ, between two points with the same ordinate on the lines $x = 0$ and $x = 1$, goes to $0$ when $y$ approaches infinity.

I need to show that lim f(n) = 0, but I do not know hot to get f(n)?

The points on $x = 0$ are $i, 2i, 3i,\ldots,$ and on $x = 1$ are $1 + i, 1 + 2i, 1 + 3i,\ldots$

Welcome to MSE. It is in your best interest that you use MathJax. — José Carlos Santos, May 26 '18 at 13:05
you should use the formula for the distance, say from here https://en.wikipedia.org/wiki/Poincar%C3%A9_metric#Metric_and_volume_element_on_the_Poincar%C3%A9_plane — orangeskid, May 26 '18 at 13:17
You are probably working in the half-plane model, but you should mention this. In other models the distances might approach infinity or be constant. — Zeno Rogue, May 26 '18 at 22:14

score 0 · Answer 1 · answered May 27 '18 at 18:42

You don't need an exact formula for the distance $f(n) = d(ni,1 + ni)$ in order to prove that the limit of the distance is equal to zero. You could instead use the squeeze theorem.

Consider the parameterized horizontal segment $$x + ni, \quad 0 \le x \le 1 $$ Its endpoints are $ni$ and $ni+1$, and therefore $d(ni,1+ni)$ is less than or equal to the length of this segment. That length is given by the path length integral $$\int_{x=0}^{x=1} \frac{\sqrt{dx^2+dy^2}}{n} = \int_{x=0}^{x=1} \frac{dx}{n} = \frac{1}{n} $$ and therefore $$0 \le d(ni,1 + ni) \le \frac{1}{n} $$ Now apply the squeeze theorem.

Show that the distance ρ, between two points with the same ordinate on the lines x = 0 and x = 1, goes to 0 when y approaches infinity.

1 Answers1