All Questions
6
questions
1
vote
1
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113
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Simple proof that $\sup\{b^t : t \in \mathbb{Q} \text{ & }t≤x\} =\sup\{b^t : t \in \mathbb{Q}\text{ & }t<x\}$
Fix $b>1$. Let $B(x) = \{b^t : t \in \mathbb{Q}\text{ & }t≤x\}$ and let $B'(x) = \{b^t : t \in \mathbb{Q}\text{ & }t<x\}$.
Show that $\sup B(x) = \sup B'(x)$. It is quite easy to show ...
- 5,154
2
votes
2
answers
3k
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If sup A $\lt$ sup B show that an element of $B$ is an upper bound of $A$
(a) If sup A < sup B, show that there exists an element of $b \in B$ that is an upper bound for $A$.
I have argued that if sup A $\lt$ sup B, then choose an $\epsilon>0$ such that sup A +$\...
- 3,286
0
votes
3
answers
62
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Can someone point out the holes in my proof using well ordering, archimedean and completeness to prove for a given set E, inf(E) = -sup(E)
Let, $$E\subset \mathbb R$$ be a non empty bounded above set. Define $$ -E = \{ -x : x \in E \}.$$ Then, $\operatorname{inf} (-E) = -\operatorname{sup}(E)$.
Proof - It follows from the completeness ...
- 319
0
votes
2
answers
1k
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How to show/proof that the union of two non empty subsets of ${\Bbb R_{}}$ has a least upper bound?
We have two sets ${E}$ and ${T}$, that are non empty subsets of ${\Bbb R_{}}$ and are bounded above.
How can I prove that,
${E}$ ${\cup}$ ${T}$ has a least upper bound (supremum), and that ${\sup(E\...
- 414
0
votes
2
answers
2k
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How can I proof the infimum and supremum of this set?
$E = \{{x+y : x,y \in\Bbb R_{>0}}$}
I was able to figure out that this set does not have a supremum, but I am not able to prove it. Also, how can I prove the infimum of this set ?
This is my ...
- 414
1
vote
2
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4k
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An upper bound $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$
Problem:
Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$.
Prove that $u$ is the supremum of $A$
if and only if for all $\epsilon > 0$ there is an $a \in A$ such that
$u-\epsilon &...
- 6,684