All Questions
Tagged with sumset supremum-and-infimum
7
questions
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2
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61
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Whether $\sup(\sum\limits_{i=1}^{\infty}X_i)$ is equal to $\sum\limits_{i=1}^{\infty}(\sup X_i)$
We have $\sup(A+B)=\sup(A)+\sup(B) $.Thus, we have $\sup(\sum\limits_{i=1}^{n}X_i)=\sum\limits_{i=1}^{n}(\sup X_{i})$ for every finite integer $n\in\mathbb{N}$. However, what about the set sequence? ...
- 117
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Following up with a previous question on $\sup(A)+\sup(B) = \sup(A + B)$
The question link is here:
Prove that $Sup(A + B) = Sup(A) + Sup(B)$
Can someone look at the answer given and explain why epsilon is introduced and how that whole second part works?
- 2,391
1
vote
1
answer
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Prove that $Sup(A + B) = Sup(A) + Sup(B)$
Earlier on in the book it showed that to prove $a = b$ it is often best to show that $a \leq b$ and that $b \leq a$. This is the way I want to go about the proof. I am sure there is an easier way but ...
- 2,391
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3
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137
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Supremum of Sumset (Proof Writing)
Given $A,b\subseteq\mathbb{R}$, define the set $A+B=\lbrace a + b | a\in A, b\in B\rbrace$. I would like to prove that $\sup(A+B)=\sup(A)+\sup(B)$, but in a specific way.
Here is what I have done so ...
- 2,125
0
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2
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1k
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Proving $\sup \left( {A + B} \right) = \sup A + \sup B$ using the usual definition.
Prove $\sup \left( {A + B} \right) = \sup A + \sup B$ using the definition given in this problem (link). I was able to prove using lemma of the given definition. My attempt is here,
let $s = \sup A$,...
user405401
1
vote
1
answer
101
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Question about Schnirelmann Density and Sumset: if $d(A) \ge \frac{1}{2}$ and $d(B) > 0$, wouldn't $d(A+B)=1$
I've been thinking about the Schnirelmann Density and I think that I may still be confused about SumSet and Density.
It seems to me that if $d(A) \ge \frac{1}{2}$ and $d(B) > 0$, then $d(A+{B}) = ...
- 10.5k
49
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5
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How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$
If $A,B$ non empty, upper bounded sets and $A+B=\{a+b\mid a\in A, b\in B\}$, how can I prove that $\sup(A+B)=\sup A+\sup B$?
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