Let $A$ and $B$ be bounded sets of real numbers such that $a\le b$ for all $a \in A$ and for all $b \in B$. Show that $\sup(A) \le \inf(B)$
Pf: Assume A and B are bounded sets. This means they have a least upper bound and a greatest lower bound.
let $x \le \sup(A)$ for some $x \in A$ and $\inf(B) \le y $ for some $y \in B$
Since we know that $a \le b $ for $\forall a\in A $ and for $\forall b \in B$, $x \le y$
Therefore, $x \le \sup(A) \le \inf(B) \le y$
Thus proving that $\sup(A) \le \inf(B)$