I am learning about Farey series from Hardy and Wright Intro to Number Theory and they give the following theorem whose proof I currently have trouble understanding.
$ \textbf{Theorem 28} $: If $ \displaystyle \frac{h}{k} $ and $ \displaystyle \frac{h'}{k'} $ are two consecutive terms of $ \mathcal{F}_{n} $, then $ h'k - hk' = 1 $.
I am reading section 3.3 First proof of Theorem 28 and the proof, extracted from the book, is given below for convenience:
$ \textbf{Proof} $: The theorem is true for $ n = 1 $; we assume it true for $ \mathcal{F}_{n - 1} $ and prove it true for $ \mathcal{F}_{n} $. Suppose that $ h / k $ and $ h' / k' $ are consecutive in $ \mathcal{F}_{n - 1} $, but separated by $ h'' / k'' $ in $ \mathcal{F}_{n} $. Let $ h''k - hk'' = r > 0 $ and $ h'k'' - h''k' = s > 0 $. Solving these equations for $ h'' $ and $ k'' $ and remembering that $ h'k - hk' = 1 $, we obtain: $ h'' = sh + rh' $ and $ k'' = sk + rk' $. Here $ \gcd(r, s) = 1 $ since $ \gcd(h'', k'') = 1 $.
Consider now the set $ S $ of all fractions $ \displaystyle \frac{H}{K} = \frac{\mu h + \lambda h'}{\mu k + \lambda k'} $ in which $ \lambda $ and $ \mu $ are positive integers and $ \gcd(\lambda, \mu) = 1 $. Thus $ h'' / k'' $ belongs to $ S $. Every fraction of $ S $ lies between $ h / k $ and $ h' / k' $ and is in its lowest form since any common divisor of $ H $ and $ K $ would divide $ k(\mu h + \lambda h') - h(\mu k + \lambda k') = \lambda $ and $ h'(\mu k + \lambda k') - k'(\mu h + \lambda h') = \mu $.
$ \textbf{Hence every fraction of $ S $ appears sooner or later in some $ \mathcal{F}_{q} $; and plainly the first to} $
$ \textbf{make its appearance is that for which $ K $ is least, i.e. that for which $ \lambda = \mu = 1 $. This} $
$ \textbf{fraction must be $ h'' / k'' $ and so $ h'' = h + h' $ and $ k'' = k + k' $.} $
I am having trouble understanding the bold paragraph. What do they mean by "Hence every fraction of $ S $ appears sooner or later in some $ \mathcal{F}_{q} $; and plainly the first to make its appearance is that for which $ K $ is least" and why is that the case. Also, why does the fraction with $ K $ least must be $ h'' / k'' $?
Any help is appreciated
Okay, to expand on the bolded paragraph:
In the preceding steps, we've shown that if $\frac{h}{k}$ and $\frac{h'}{k'}$ are consecutive in $\mathcal F_n$ but not in $\mathcal F_{n+1}$, then any fraction $\frac{h''}{k''}$ that lies between them in $\mathcal F_{n+1}$ has to be of the form $\frac{h''}{k''} = \frac{\mu h + \lambda h'}{\mu k + \lambda k'}$.
So let $S = \left\{\frac{\mu h + \lambda h'}{\mu k + \lambda k'} : \lambda, \mu \ge 1, \gcd(\lambda,\mu) = 1\right\}$: the set of all things that $\frac{h''}{k''}$ could possibly be. Each element of $S$ is a fraction in simplest terms, so each element of $S$ appears in some Farey series. For example, $\frac{2h + 3h'}{2k + 3k'}$ appears in the Farey series $\mathcal F_{2k+3k'}$, by definition.
We know that none of the fractions in $S$ have appeared in $\mathcal F_n$ yet: if they did, they would be between $\frac hk$ and $\frac{h'}{k'}$, but we are told that $\frac hk$ and $\frac{h'}{k'}$ are consecutive in $\mathcal F_n$. On the other hand, at least one of the fractions in $S$ appears in $\mathcal F_{n+1}$: it's $\frac{h''}{k''}$. Therefore no fraction in $S$ has denominator $n$ or less, and $\frac{h''}{k''}$ is in $S$ and has denominator $n+1$ or less. It must have denominator exactly $n+1$, and this must be the least possible denominator of any fraction in $S$.
The least possible denominator of a fraction of the form $\frac{\mu h + \lambda h'}{\mu k + \lambda k'}$, where $\mu,\lambda \ge 1$, is obtained when $\mu = \lambda = 1$. So $\frac{h + h'}{k + k'}$ is the (unique) fraction in $S$ with the least denominator, and therefore $\frac{h''}{k''} = \frac{h+h'}{k+k'}$.