Observe that I is a set of natural numbers. If I∈U, then I cannot be finite, and it seems pretty obvious that almost all the elements in a,b are the same (they only disagree at a finite number of places after all).
The bracketed remark doesn't appear to be true. Why can we not have I={0,2,4,...}∈U or I={1,3,5,...}∈U? Indeed, by the definition of an ultrafilter, we must have one of them in U. Also, in the post, you use I for two different purposes, which makes the post slightly less clear.
The bracketed remark doesn't appear to be true. Why can we not have I={0,2,4,...}∈U or I={1,3,5,...}∈U? Indeed, by the definition of an ultrafilter, we must have one of them in U. Also, in the post, you use I for two different purposes, which makes the post slightly less clear.