Taken from A Course of Pure Mathematics:
Irrational Numbers: If the reader will mark off on the line all the points corresponding to the rational numbers whose denominators are $1,2,3\ldots$ in succession, he will readily convince himself that he can cover the line with rational points as closely as he likes. We can state this more precisely as follows: if we take any segment $BC$ on $A$, we can find as many rational points as we please on $BC$.
Suppose, for example, that $BC$ falls within the segment $A_1A_2$. It is evident that if we choose a positive integer $k$ so that$$k.BC>1\tag1$$ And divide $A_1A_2$ into $k$ equal parts, then at least one of the points of division (say $P$) must fall inside $BC$, without coinciding with either $B$ or $C$. For if this were not so, $BC$ would be entirely included in one of the $k$ parts into which $A_1A_2$ has been divided, which contradicts the supposition $(1)$. But $P$ obviously corresponds to a rational number whose denominator is $k$. Thus at least one rational point $P$ lies between $B$ and $C$. But when we can find another such point $Q$ between $B$ and $P$, another between $B$ and $Q$, and so on indefinitely; i.e., as we asserted above, we can find as many as we please. We may express this by saying that $BC$ includes infinitely many rational points.
Question: What does the passage mean by this? I can't follow through with the proof, can you explain it to me?
I tried drawing a diagram with horizontal line $A_1A_2$ and starting off with an example, such as when $BC=1/3$ and $k=4$. However, no matter what, I just can't follow through.
The English is confusing me.