prove or disprove: sup of two sets

**Stormey** · January 21st 2013, 04:48 AM

Hi,
i need to prove or disprove the following:

$Sup(A\cup B)=max\left \{ SupA, SupB \right \}$

thanks in advanced!

*edit:
both sets are bounded from above.

**emakarov** · January 21st 2013, 05:10 AM

Prove (1) $\sup(A\cup B)\le\max(\sup A,\sup B)$ and (2) $\sup(A\cup B)\ge\max(\sup A,\sup B)$ .

For (1), show that $\max(\sup A,\sup B)$ is an upper bound of $A\cup B$ .

For (2), use the fact that $x\ge y$ and $x\ge z$ imply $x\ge\max(y,z)$ .

**johng** · January 21st 2013, 07:53 AM

I've found the following to be useful:
supremums of larger sets are larger and infimums of larger sets are smaller

Formally, (assuming the sets are bounded):
1. If $C\subseteq D$ then $sup(C)\leq sup(D)$
2. If $C\subseteq D$ then $inf(C)\geq inf(D)$

Proof of 1 - Let $u=sup(D)$ . Let $x\in C$ , then $x\in D$ and so $x\leq u$ ; i.e. $u$ is an upper bound of $C$ . So $sup(C)\leq u$ . The proof for infs is exactly similar.

For your original question, since $A\subseteq A\cup B$ , $sup(A)\leq sup(A\cup B)$ , and similarly for B.

**Stormey** · January 21st 2013, 09:06 AM

Thanks guys.

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