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**Gusbob** $\displaystyle a_2 $ = gcd $\displaystyle ( a_1, b_1 ) $ implies that $\displaystyle a_2=r\cdot a_1$ for some $\displaystyle r\in R$, so $\displaystyle \langle a_1\rangle \subseteq \langle a_2 \rangle$. Now $\displaystyle \langle a_2 \rangle \subseteq I$ by definition, so the question becomes whether $\displaystyle \langle a_2 \rangle$ contains all elements of $\displaystyle I$, formulated as $\displaystyle I=\langle a_2 \rangle$ or not.

The question of `$\displaystyle I=\langle a_i \rangle$?' is always valid, the motivation being the gcd bit, which guarantees $\displaystyle \langle a_{i-1}\rangle \subseteq \langle a_i \rangle$. What you are doing in this proof is constructing bigger and bigger principal sub-ideals $\displaystyle \langle a_i\rangle$ of $\displaystyle I$, and argue that you must eventually get all of $\displaystyle I$ (otherwise you get a contradiction). Essentially, the reason we want to ask whether $\displaystyle I=\langle a_i \rangle$ 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.