TITLE: Is $1$ the halfway point between $0$ and infinity?
QUESTION [13 upvotes]: I am a math newbie, but I stumbled across this while I was working on a game in python. This is the proof I have, I believe it shows that the halfway point between $0$ and infinity is $1$:

If it is true, I'm probably not the first to figure this out, but even still, I just want some kind of confirmation.
This is very mind blowing for me right now :D

REPLY [2 votes]: Another notion where 1 could be seen as "halfway" between 0 and +infinity is the Farey sequence:
$$  F_1 = { \frac 0 1, \frac 1 1 } $$
  $$  F_2 = { \frac 0 1, \frac 1 2, \frac 1 1 } $$
  $$  F_3 = { \frac 0 1, \frac 1 3, \frac 1 2, \frac 2 3, \frac 1 1 } $$
If we try to extend it from $(0,1)$ to $(0, +\infty)$, the simplest solution is to start from:
$$  E_0 = { \frac 0 1, \frac 1 0 } $$
and we get the extended sequence, where every term has all the fractions of the original term and their symmetrical fractions::
  $$  E_1 = { \frac 0 1, \frac 1 1, \frac 1 0 } $$
  $$  E_2 = { \frac 0 1, \frac 1 2, \frac 1 1, \frac 2 1, \frac 1 0 } $$
  $$  E_3 = { \frac 0 1, \frac 1 3, \frac 1 2, \frac 2 3, \frac 1 1, \frac 3 2, \frac 2 1, \frac 3 1, \frac 1 0 } $$