I am self-teaching myself through the Concrete Mathematics book, and recently saw the proof for (no rational numbers are omitted in stern brocot tree construction) as mentioned in link below :
The part that I have trouble understanding is the line above to below with "imply that",
I believe $\frac{a}{b} - \frac{m}{n} \gt 0$ should lead to $an - bm > 0$
but why/how does it imply in the proof
that $an - bm \ge 1$ ?
Hint (expanding on the previously posted comment): in integers $a \gt b \iff a \ge b+1\,$.
The above is equivalent to $n \gt 0 \iff n \ge 1$ which follows from the well-ordering principle. The set $X = \{n \in \mathbb{Z} \mid n \gt 0\}$ must contain a least element. Since $0 \not\in X$ and its successor $1 \in X$ it follows that the least element is $1\,$, therefore $n \ge 1\,$. (If this argument is not satisfactory then you'll need to specify the particular formalism/construction of integers that you are working with.)