I have been reading "Prelude to Mathematics" by W.W. Sawyer. In it he describes Poincare's hypothetical universe.
This universe is contained in the interior of a circle. The temperature of this universe is high in the centre and gets cooler as you move towards the circumference. The law for the temperature is: If a is the radius of the circle, at the distance r from the centre of the circle the temperature is T= aČ - rČ.
The size of objects inside this universe are affected by the variation in temperature. The length of any object varies in proportion to the temperature T. At the circumference of the circle, where r=a and T=0, the length of an object will shrink to zero.
Inhabitants of this universe are unaware of the changing temperature as they have no sensitivity to it. They are also unaware of their change in size, as anything they use to measure their size also changes its size.
The people/creatures inside this universe can never get to the boundary because the nearer they get to the boundary the more rapidly they shrink and the smaller their steps become. So although the universe is finite and bounded to us, to the inhabitants it is infinite.
This all makes perfect sense to me. The only part I do not understand is Sawyer's proof using simple calculus that the inhabitants can never reach the boundary (or circumference) of their universe. I studied basic calculus 20 years ago but I am a bit rusty and cannot follow Sawyer's calculation pasted below. Please can someone explain the last part of this calculation to me? Is it possible to break it down into simpler steps, or at least explain how the last calculation is arrived at? I would be most grateful for your help. Rob