proof of fixed points of normal functions

Suppose f is a κ-normal function and consider any α<κ and define a sequence by α0=α and αn+1=f(αn). Let αω=supn<ωαn. Then, since f is continuous,


So Fix(f) is unboundedPlanetmathPlanetmath.

Suppose N is a set of fixed pointsPlanetmathPlanetmath of f with |N|<κ. Then


so supN is also a fixed point of f, and therefore Fix(f) is closed.

Title proof of fixed points of normal functions
Canonical name ProofOfFixedPointsOfNormalFunctions
Date of creation 2013-03-22 13:29:01
Last modified on 2013-03-22 13:29:01
Owner Henry (455)
Last modified by Henry (455)
Numerical id 4
Author Henry (455)
Entry type Proof
Classification msc 03E10