Look for the comment in the article, after passing to a subsequence if necessary. The ultrafilter produces the necessary subsequence for any question that you ask, and will do so in such a way as to produce logically consistent answers for any combination of questions that you choose.
That is why the ultrafilter axiom is a weak version of choice. Take the set of possible yes/no questions that we can ask as predicates, such that each answer shows up infinitely often. The ultrafilter results in an arbitrary yet consistent set of choices of yes/no for each predicate.
The axioms demand that either one function is eventually dominated by the other, or both functions are of the same order. But which of these is the case will strongly depend on which subsequence you look at.
You may have missed the same subtlety that I did. Because pi is irrational, the functions are different at all integers. Therefore, in the total order, these two functions cannot have the same order.
That still doesn't resolve which one is larger though.
Well, as presented in Tao's post, the set Ω can be either the natural numbers or the real numbers. So I'm assuming the "subsequence" is a (perhaps uncountable?) set of real parameters, in the latter case.
