The problem statement does not say that b >= a. If a > b, then q = 0 and r = b.

Are m and n integer? Why do they exist?Let r=\frac{mb}{n} where m<n<2m

I have not read your proof to the end, but I think there is an easier direct proof. Suppose b >= a. If a <= b/2, then r < a <= b/2. If a > b/2, then q = 1 and r = b - a < b/2.