Brief refresher on the fast growing hierarchy for finite subscripts:
where we use function iteration notation. It may easily be derived that
which are left to the readers as induction exercises.
I then define tetration with a modified top exponent as follows:
I shall then prove the following bounds for all :
We do this by seeing it holds for and then applying induction over with in the following:
Trivially it is true for . Assume it holds for some arbitrary , and we shall prove it holds for .