If $m$ and $n$ are coprime integers, prove that each of these $m+n2$ fractions: $$\frac{m+n}{m},\frac{2(m+n)}{m},\frac{3(m+n)}{m},...,\frac{(m1)(m+n)}{m},$$ $$\frac{m+n}{n},\frac{2(m+n)}{n},\frac{3(m+n)}{n},...,\frac{(n1)(m+n)}{n}$$ fits one of these open intervals $(1,2),(2,3),(3,4),...,(m+n2,m+n1)$
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