Supremum and Infimum: Denote B={|x-y|;x∈A and y∈A}....

Alfredo Dawlabany

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Let A be a non-empty bounded set of \(\displaystyle \mathbb{R}\). Denote B={|x-y|;\(\displaystyle x\in A\) and \(\displaystyle y\in A\)}.
Prove that Sup B = Sup A - Inf A.

I tried to solve it but didn't know what to do

I said that if A is a bounded set then \(\displaystyle A\subset \left [InfA,SupA\right ]\)
\(\displaystyle \Rightarrow x,y \in \left [ InfA,SupA \right ]\)
so we have \(\displaystyle InfA\leqslant x\leqslant SupA\) and \(\displaystyle InfA\leqslant y\leqslant SupA\)


Am I going right ? If yes how would I continue ?
 

tkhunny

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Just hints:

1) Does it matter or help that Sup and Inf are actually members of the sets?
2) I don't see any subtraction in your demonstration, so far. What's up with that?
 
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Alfredo Dawlabany

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Just hints:

1) Does it matter or help that Sup and Inf are actually members of the sets?
2) I don't see any subtraction in your demonstration, so far. What's up with that?
What do you mean by subtraction ?
 

tkhunny

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|x-y|
 

Alfredo Dawlabany

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OK I understood
But there is the absolute value
-sup A≤-y≤-inf A
So we get
x-y≤sup A - inf A , this implies sup(x-y)=sup A - inf A
Then what?
 

tkhunny

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|x-y| is just distance. You will lose no generality if you assume x > y. Obviously, if x = y, we have a minimum, but we're not looking for that.

Rephrasing, how far apart can x and y be?
 

Alfredo Dawlabany

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|x-y| is just distance. You will lose no generality if you assume x > y. Obviously, if x = y, we have a minimum, but we're not looking for that.

Rephrasing, how far apart can x and y be?
Okay
I understood everything
Thank you
 
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