How to prove $a \lt b$ for all $a \in A, b \in B$ implies $\sup A \leq \inf B$? If $A, B$ are two non empty sets of real numbers and for every $a$ from $A$ and $b$ from $B$

$a < b$,

how can I prove that 

$\sup A \leq \inf B$?


I don't think this is true. Let $A=(0,1)$ and $B=(1,2)$ then $\operatorname{sup}A=1$ and $\operatorname{inf}B=1$ but $A$ and $B$ satisfy your conditions.
