All of SatvikBeri's Comments + Replies

#6:

Assume WLOG $f\left(x\right)>=x$Then by monotonicity, we have $x<=f\left(x\right)<=f^2\left(x\right)<=...<=f^{|P|}\left(x\right)$If this chain were all strictly greater, than we would have $|P|+1$istinct elements. Thus there must be some $k$uch that $f^k\left(x\right)=f^{k+1}\left(x\right)$By induction, $f^{n+1}\left(x\right)=f^n\left(x\right)=f^k\left(x\right)$or all $n>k$

#7:

Assume $f\left(x\right)>=x$nd construct a chain similarly to (6), indexed by elements of $\alpha$If all inequalities were strict, we would have an injection from $\alpha$o L.

#8:

Let F be the set of fixed points. Any subset S of F must have a least upper bound $x$n L. If x is a fix... (read more)