# Thread: Some properties on supermum

1. ## Some properties on supermum

T or F:

Suppose that A and B are bounded sets in the real numbers.

1) sup (A U B) = max {sup A, sup B}

2) If the elements of A and B are positive and A.B = {ab | a is in A, b is in B}, then sup(A.B) = sup(A)sup(B)

3) The analogous problems for the greatest lower bound.

I believe all of them are true, but just wanted to make sure before I started on the proofs. Thanks!

2. Hi.

Let $\displaystyle C$ be a bounded real set, the set of upper bounds of $\displaystyle C$ has always a minimal element, which is the supremum of $\displaystyle C.$

Now for instance, 1):
$\displaystyle A,B$ two bounded real sets.
$\displaystyle a=supA$ and $\displaystyle b=supB$ and $\displaystyle c=sup(A\cup B).\ \forall t\in \mathbb{R},$

$\displaystyle t=c$
$\displaystyle \Leftrightarrow (\forall x\in A\cup B, x\leq c)\wedge (\forall y\in\mathbb{R}(\forall x\in A\cup B,\ y\geq x\Rightarrow y\geq c))$
$\displaystyle \Leftrightarrow (\forall x\in A,\ x\leq c\wedge \forall x\in B, x\leq c)\wedge (\forall x \in\mathbb{R} (x\geq a\wedge x\geq b)\Rightarrow x\geq c)$
$\displaystyle \Rightarrow (a\leq c\wedge b\leq c)\wedge (c\leq max\{a,b\})$
$\displaystyle \Rightarrow (max\{a,b\}\leq c)\wedge (c\leq max\{a,b\})$
$\displaystyle \Rightarrow c=max\{a,b\}$