However, there are some proofs in set theory in which this does not hold.

For example, use transfinite induction on only natural numbers.

So, we seek the least index n such that .9(n) = 1. We find it cannot be decided since for all indexes n, .9(n) != 1. This is a problem.

We also also can take an approach of considering a neighborhood around 1. We also see for all n, we can find a number between .9(n) and 1.

We can't find any natural number in which this relation is false. So, if we could actually exhaust all n, we are left in the state there is a number between .9(n) and 1. This actually means we cannot exhaust all n.

I hope we can see discussion on this issue.