Rationals: dense ordering question

Question

I'm reading about construction of $\mathbb{Q}$ on Wiki and dense ordering on Wiki.

In the former:

$(\frac a b < \frac c d) \implies (\frac a b < \frac {a + c} {b + d} < \frac c d)$

Is a theorem that establishes this? How adding numerators and denominators yields a fraction in between.

Answer 1

You can do this just by using arithmetic. For example, $a(b+d) < b(a+c)$ because $ab+ad < ab+bc$ because $ad<bc$.