Hi,

i need to prove or disprove the following:

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

thanks in advanced!

*edit:

both sets are bounded from above.

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- Jan 21st 2013, 04:48 AMStormeyprove or disprove: sup of two sets
Hi,

i need to prove or disprove the following:

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

thanks in advanced!

*edit:

both sets are bounded from above. - Jan 21st 2013, 05:10 AMemakarovRe: prove or disprove: sup of two sets
Prove (1) $\displaystyle \sup(A\cup B)\le\max(\sup A,\sup B)$ and (2) $\displaystyle \sup(A\cup B)\ge\max(\sup A,\sup B)$.

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

For (2), use the fact that $\displaystyle x\ge y$ and $\displaystyle x\ge z$ imply $\displaystyle x\ge\max(y,z)$. - Jan 21st 2013, 07:53 AMjohngRe: prove or disprove: sup of two sets
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 $\displaystyle C\subseteq D$ then $\displaystyle sup(C)\leq sup(D)$

2. If $\displaystyle C\subseteq D$ then $\displaystyle inf(C)\geq inf(D)$

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

For your original question, since $\displaystyle A\subseteq A\cup B$, $\displaystyle sup(A)\leq sup(A\cup B)$, and similarly for B. - Jan 21st 2013, 09:06 AMStormeyRe: prove or disprove: sup of two sets
Thanks guys.