Gurkenglas comments on Iteration Fixed Point Exercises

[Gurkenglas]

#3:

$x\to \frac{x+1}{x}$ on $x\ge 1$ shortens all distances but is strictly monotonic.

#6: (the “show that if” condition follows from the property, the question is likely misstated)

The iteration is so long that it must visit an element twice. We can’t have a cycle in the order so the repetition must be immediate.

[Scott Garrabrant]

Thanks, I actually wanted to get rid of the earlier condition that $f\left(x\right)\ge x$ for all $x$, and I did that.