= gcd

implies that

for some

, so

. Now

by definition, so the question becomes whether

contains all elements of

, formulated as

or not.

The question of `

?' is always valid, the motivation being the gcd bit, which guarantees

. What you are doing in this proof is constructing bigger and bigger principal sub-ideals

of

, and argue that you must eventually get all of

(otherwise you get a contradiction). Essentially, the reason we want to ask whether

at each step is because it is a principal ideal that is bigger (and contains) all previous principal ideals we looked at.

I hope that made sense, I'm not able to explain this eloquently.