In reply to

http://math.eretrandre.org/tetrationforu...73#pid4073

First it is easy to see that for :

( is the lower fixed point of )

Hence for we have for all :

(*)

We also know that for , quite fast, particularly for each there is an such that for all :

(**) .

Now we lead proof by contradiction, suppose that

where .

Then there must be a subsequence and such that this subsequence stays always more than apart from :

.

I.e. there is and such that

either or .

By (*) and (**) we have such that for all :

and .

As is monotone increasing for we have also

and .

This particularly means and hence none of the can be the self superroot, in contradiction to our assumption.

http://math.eretrandre.org/tetrationforu...73#pid4073

First it is easy to see that for :

( is the lower fixed point of )

Hence for we have for all :

(*)

We also know that for , quite fast, particularly for each there is an such that for all :

(**) .

Now we lead proof by contradiction, suppose that

where .

Then there must be a subsequence and such that this subsequence stays always more than apart from :

.

I.e. there is and such that

either or .

By (*) and (**) we have such that for all :

and .

As is monotone increasing for we have also

and .

This particularly means and hence none of the can be the self superroot, in contradiction to our assumption.