¿Has the proof of Theorem 1.21 of Principles of Mathematical Analysis by Walter Rudin a meaning?

233 Views Asked by At

I did the proof of this theorem a slightly different some time ago, but, I really forgot almost everything and the problem is that, I understand the body of the proof, but, I don't understand why he use those numbers and inequalities (Despite on the fact that they works). I think is the same for some other proofs of the book.

My question is about, how much background has the proof?, to get those expressions

Here is the proof

enter image description here

enter image description here