I am reading a research paper on 'Factoring Algorithms for Quantum Computation' by Peter W. Shor. In this paper there is written somewhere :-
(n,c,d,q,r are all positive integers)
$$\frac{-r}{2} \le rc-dq \le \frac{r}{2} \tag{6.9}$$ Dividing by rq and rearranging the terms gives $$\left|\frac{c}{q}-\frac{d}{r}\right|\le \frac{1}{2q} \tag{6.10}$$ We know $c$ and $q$. Because $q \ge 2n^2$, there is at most one fraction $d/r$ with $r < n$ that satisfies the above inequality.
Here i didn't understand that what is meant by only one $d/r$, why it can not be more than one.
My approach:-
Let $q=2n^2$ and $r=n-a$ then eq. $(6.10)$ becomes
$$\left|\frac{c}{2n^2}-\frac{d}{n-a}\right|\le\frac{1}{4n^2}$$
Now since $$\frac{1}{2n^2}=2\times \frac{1}{4n^2}$$ so $d/(n-a)$ has to be $(1/4n^2)+something$
From here things got messy, and i started feeling that I'm going in the wrong direction.I want to understand why there will be only one fraction $d/r$ satisfying the foretold inequality?
Please Help !