It seems like you have become distracted from the topic of this thread, OP paper, by some of the pictures that have been posted. OP paper has every article item specified by a three digit unique ID but you don't cite any such unique ID in your question. Why?>So how do you justify that?
Well, if you're talking about the picture of the "Zeros" paper which you linked, then my justification is that that paper is about the treatment of RZF with INF_HAT and not INF_HAT itself. I do the formal treatment of INF_HAT in OP paper. I give the full construction of R by Cauchy sequences and Dedekind cuts therein. I give the arithmetic axioms and prove their well-definition. >So how do you justify that?
You mention something about INF and INF_HAT both being equal to the same limit which does not appear in the "Zeros" paper at all. You must be talking about a paper different than one you linked, I presume, because you are trying to confuse the line of your reasoning by mixing up different pictures of papers and not citing the articles in any of them by their unique IDs. If you have also gotten distracted away from OP paper by the picture of the "Quick" paper, which does use the limit definition that you cite, then you should know that the "Quick" paper is not about supporting Prop 1.8. That paper is about the consequences which follow if I assume that Prop 1.8 is true. In OP paper, in which I give the formal treatment of INF_HAT, I make the distinction
as the workaround for your question about how two equal things might behave differently. If you continue your inquiry, please cite a three number article identifier from OP paper, e.g: Thm X.Y.Z, so I know you are not getting distracted by the pictures. If you are more interested in the pictures, however, and you want to ignore OP paper that answers all the questions you will ask, then please cite the Title and two digit article ID.