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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 .

Q.E.D.

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