Ordered field: If $a>0$, $b>0$, $c>0$, $d>0$, $a<b$, $d<c$, do we have $ad<bc$?

Question

In an ordered field, if we should have $a>0$, $b>0$, $c>0$, $d>0$, $a<b$, $d<c$, do we have $ad<bc$?

Please include the proof/counterexample.

Answer 1

In an ordered field, if < and >0, then <.

So:

$a<b$

$d>0$

$=> ad<bd$

$d<c$

$b>0$

$=> bd<bc$

$=> ad<bc$