I've been experimenting looking at these operators and I've become much more familiar with its structure. I can prove convergence now.

Let's suppose by contradiction that; for sufficiently large n:

However. It is clear that for some

1:

Therefore we write:

Now we know that

Therefore:

However this is a contradiction because for sufficiently large the left equation becomes much larger than the right. This is easy to deduce by the relation 1 above. Therefore to prove convergence we just need add the claim:

YES! We have convergence for all s.

An important theorem I have to prove is the following, I consider it a stern requirement of hyperoperators. For all and and

I'll mull over that for awhile.

also; hopefully:

for at least

Let's suppose by contradiction that; for sufficiently large n:

However. It is clear that for some

1:

Therefore we write:

Now we know that

Therefore:

However this is a contradiction because for sufficiently large the left equation becomes much larger than the right. This is easy to deduce by the relation 1 above. Therefore to prove convergence we just need add the claim:

YES! We have convergence for all s.

An important theorem I have to prove is the following, I consider it a stern requirement of hyperoperators. For all and and

I'll mull over that for awhile.

also; hopefully:

for at least