Suppose I am using Shor's order finding algorithm to calculate the order $r$ of $x\leq N$ with respect to $N$. After some run of the QPE subroutine, I obtain a good, $L$-bit approximation to $s/r$ for some $s\leq r$. According to, say, Nielsen and Chuang, because my estimation $\phi$ is sufficiently close, when I do the continued-fractions algorithm (CFA) on $\phi$, one of the convergents will be exactly $s/r$.
My question is, how will we know which convergent? It certainly isn't necessarily the last convergent, since this will be equal to the binary approximation $\phi$ and not $s/r$. Is there a guess-and-check stage that goes on here and if so, does efficiently of the algorithm depend on the number of convergents being small relative to $N$?
Edit: I should mention that I am also curious why the answer is what it is. Namely, if we are able to make a specific choice of convergent, what mathematical argument allows us to do so?