Idea behind proof of Division Algorithm

Question

Why did they take set S as they have taken it ?

Why did they say to complete proof we have to show that r < b?

Answer 1

Why did they take $S$ as they have taken it?

They want to show that given some fixed $a,b\in\mathbb{N}$ you can find $q,r\in\mathbb{N}$ such that $a=qb+r$. This criterium is the same as saying $a-qb=r$. So we could let $q$ run over all positive integers, and for each $q$ we will get an $r$. Now, we want to find the smallest $r$ possible.

Why did they say to complete proof we have to show that $r < b$?

If this is not the case then we can always make $q$ bigger, that is, subtracting another $b$, since $b>r$, so without loss of generality it's safe to assume that $r<b$ (otherwise $r=0$).

Answer 2

The goal is to prove that, given natural numbers $a$ and $b$, there are non-negative integers $q$ and $r$ such that $a=bq+r$, with $0\leqslant r<b$. So, it is natural to study the numbers of the form $a-bt$ to prove this ($a=bq+r\iff a-bq=r$).

And after haveing proved that you can write $a$ as $bq+r$ for some non-negative integers $q$ and $r$, you still must prove that $r<b$, because that's part of the statement of the theorem.