On proof methodology of Theorem 1.20(b) in Rudin's "Principles of Mathematical Analysis," 3rd edition. I am studying Rudin's "Principles of Mathematical Analysis," 3rd edition by myself. On reading a few proofs in the book I am getting an idea of the following methodology in proving theorems :

*

*Ask a question. For example, "Is $\mathbb{Q}$ dense in $\mathbb{R}$?"


*State what needs to be proved if the answer to the above question is true. In our case,

If $ \in \mathbb{R}$, $y \in \mathbb{R}$, and $<$, then there exists a $p \in \mathbb{Q}$ such that $<<$

If this proposition is disproved then the answer to 1 is NO, otherwise YES


*Work backwards and try to find what need to be proved from which 2 can be deduced. In our case,

$\exists  m, n \in \mathbb{Z}, n \ne0: nx < m < ny$

Here we simply applied the definition of a rational number.


*Continue this process until you reach a proposition which can be proved/disproved using another theorem or axiom. In our case, Rudin has used the Archimedean Property of $\mathbb{R}$


*Write the steps from 4 to 2 (in that order). This completes the proof/disproof.
This methodology appears far convincing to me when compared to knowing the reason behind : "Why Rudin used $n(y-x) > 1$ in the first place?"
I would like to know if this is the right methodology?
 A: This is correct - as far as it goes. But your question

Why Rudin used $n(y−x)>1$ in the first place?

is in a way the crux of the matter. The other steps are the routine part - the framework, as it were. This is the place where you need some actual understanding/intuition about "why" the assertion is correct. That you develop over time, by looking at lots of examples and lots of proofs, trying to reconstruct the ideas, not just check the logic.
Good for you for understanding that when you write the formal proof you write those steps in reverse order - not the way you discovered them.
In "prove or disprove" questions (the most instructive kind) you often find yourself first trying to prove something. If you run into a wall you wonder whether the assertion is true, so you start looking for a counterexample. If you can't find one you go back to looking for a proof. That back and forth continues until you understand the situation.
Finally, an anecdote. Years ago I took advanced calculus from  Lars Ahlfors. He warned us that he did not like proofs that started from the hypotheses and got as far as possible, then from the conclusion reasoning backwards as far as possible, then writing "therefore" where the logical chains met.
