An example of a fixed point iteration not converging to a fixed point in one countable sequence
Start from e.g. x=1/2. Obtain a series converging to 1, but 1 is not a fixed point because .
This is because is not continuous. If it was continous (at least, continuous from the left), we would have the desired property.
Whenever we converge to some , we take and continue iterating and taking limits (the “number of times” to iterate, even if infinite, depends on the size of the lattice, which can be formalized ordinal numbers).