Construction of Sequence of Numbers with Given Ratios

Proof
Let the given ratios in least numbers be:
 * $a : b$, $c : d$, $e : f$

From, we can find:
 * $g = \operatorname{lcm}\left\{{b, c}\right\}$

Let:
 * $g = r b$
 * $g = s c$

Then let:
 * $h = r a$
 * $k = s d$

Now either $e \mathop \backslash k$ or $e \nmid k$.

First, suppose $e \mathop \backslash k$.