"But I think it is wrong. From x^x^x... convergent, you cannot get x=x^x. Let's say x=.5. .5, .5^.5, .5^.5^.5,... is convergent because it is increase and smaller than 1^.5^.5..., which is 1. But you cannot say .5=.5^.5."
your .5=.5^.5 is wrong, should be x=x^x, what is x, x is the limit, it is not .5.
let me tell you the detail,(not real Math):
x= .5^.5^.5^.5^.......^.5^.5^.5^.5^.5^......=x^x.
This is not strict Math,but good for understanding.