Clarification for a simple argument in an inequality Suppose $a,b,c,d \in \mathbb{R}^+$ are such that $$ a \leq b \\ c \leq d \\ ac = bd $$ then is the following true? $$ a= b \\ c = d $$
Attempt:
Suppose not true, i.e. $a < b$ or $c < d$, then we have $ac < bd$ thus contradicting the fact that $ac=bd$. Thus $a= b$ and  $c = d$.
So, is my argument correct? Sorry for posting this if this is a trivial fact.

Okay, FINAL final answer:
If $0 \le a \le b$ and $0 \le c \le d$ then 
$ac \le bc$ with equality holding only if $a = b$ or $c = 0$.
And $bc \le bd$ with equality holding only if $c = d$ or $b= 0$.
So $ac \le bd$ with equality holding only if one of the four occur:
1) $a = b$ and $c = d$
2) $a =b$ and $b = 0$ so $a = b  =0$.
3) $c = 0$ and $c= d$ so $c = d = 0$.
4) $c =0$ and $b= 0$ and $0 \le a \le b = 0$ so $a = b = c = 0$
So if $ac = bd$ then either i) $a=b$ and $c=d$ or ii) $a=b=0$ or $c=d=0$.
And if $ac = bd \ne 0$ then $a=b$ and $c=d$.
Otherwise $ac < bd$.
