Whoops. Page 47. I paraphrase: Sequences "converge" if there exists an N such that the sequence stays within epsilon of p (the mark) for all indexes larger than

Whoops. Page 47. I paraphrase: Sequences "converge" if there exists an N such that the sequence stays within epsilon of p (the mark) for all indexes larger than N. I know it seems formal because it adheres to a certain structure, but even this is informal at a fundamental level. Specifically, how can we be sure we can perform a countably infinite number of distance measurements in the metric space to be sure the sequence stays close to p? (this is the infinite stack of inequalities I alluded to) He doesn't say. Implicitly, Rudin is saying here that this definition of "converges" is good enough. And he's not wrong. It is a very good definition. To me at least this is Rudin, the towering statue of formality, being informal. He could/should have actually gone down to a more fundamental level and whipped out mathematical induction as an axiom to assure us that we can do such things, but then that would have taken him off his narrative goal, and also probably lost even more readers. Furthermore, even if he did so, an axiom is an assertion that "you just have to trust me on this one." Now look, I'm not bashing formality. I'm a huge fan of it, and teach upper level math classes formally. But it has it's place and it is NOT in Calculus 1. Furthermore, I think folks need to realize that even the most formal of treatises have informalities buried in them at the very least in the form of stated axioms.