Question

Using the least integer principle, define r to be least integer for which j - qk is positive. Prove that 0< r <=k

Answer 1

? this shuold that q or j be an r?

Answer 2

Actually r= j - qk. This comes form Euclid's division lemma: j=qk + r

Answer 3

Is j and q positive or not?

Answer 4

Proofwiki has a rigorous but rather hard to understand proof. Ask if you need help.

https://proofwiki.org/wiki/Division_Theorem

Answer 5

I think I understood the informal proof. Just to make sure, when it says that:

b∖(r−r0).
−b<r−r0<b.
Hence, r−r0=0

r-r0 must be 0 because there is no other way it is divisible by b. Am I right?