Is there a formal way of saying a<b and c<d implies ac<bd, like a theorem from algebra or something? Just wondering because I used it for my inductive step.
I believe the way you wrote it should be fine for the proof.
There is a property of inequalities that states:
if c is some positive number and a < b, then ac < bc
if c is some negative number and a < b, then ac > bc
From this property we can prove inequalities such as the following:
If a<b and c<d, where a, b, c, and d are positive numbers then ac<bd must be true.
