Need hints (advice) to prove $(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$

Question

I'm trying to prove this ( source : my uni's textbook says that it's trivial).

$$(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$$

So far, I've managed to get to the point where

1. $$ab < bc \wedge bc < bd \Rightarrow ab < bd$$
2. $$ ac < bc \wedge ac < ad$$ 3 (using 2.) $$ ad < bd \Rightarrow ac < ad < bd$$

And I sort of don't know how to use this information to get to the desired result. I feel like I'm going around in circles.

Answer 1

Hint : $$ 0<(b-a)(d-c)=bd-ad-bc+ac $$