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I'm having a problem with this theorem. What if set $B$ is all $x$ such that $\sqrt{2} < x \le 2$, and $S$ is the set of all $y$ such that $\sqrt{2} < y \le 3$. $\sup L = \sqrt{2}$, which does not exist in $S$. Does this prove the theorem wrong by contradiction?


1.11 Theorem Suppose $S$ is an ordered set with the least-upper-bound property, $B \subset S$, $B$ is not empty, and $B$ is bounded below. Let $L$ be the set of all lower bounds of $B$. Then $\alpha = \sup L$ exists in $S$, and $\alpha = \inf B$.

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    $\begingroup$ It would be better to state the theorem in this space. Many viewers, including some who could give very good answers, will not have the book at hand. $\endgroup$ – Dylan Moreland Jun 6 '12 at 2:21
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    $\begingroup$ Related: math.stackexchange.com/questions/102874/rudin-theorem-1-11 $\endgroup$ – Jonas Meyer Jun 6 '12 at 2:23
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    $\begingroup$ ... and I do have the book, but apparently a different edition (mine is Second Edition), because Theorem 1.11 in mine is about the $<$ relation on Dedekind cuts. $\endgroup$ – Robert Israel Jun 6 '12 at 2:27
  • $\begingroup$ There is a exercise in Rudin related to this, although I forget what chapter. $\endgroup$ – zzzzzzzzzzz Jun 6 '12 at 14:52
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No, your example does not meet the hypotheses of the theorem, because $B$ is not bounded below in $S$. Note that $L=\{a\in S:\text{for all }b\in B, a\leq b\}=\emptyset$.

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  • $\begingroup$ Is there a reason why you make some of your answers community wiki? $\endgroup$ – user17762 Jun 6 '12 at 2:26
  • $\begingroup$ @Marvis: Sometimes I feel like doing so. This is more common when I have very little to say. Others are welcome to edit. $\endgroup$ – Jonas Meyer Jun 6 '12 at 2:30
  • $\begingroup$ First, what is "bounded below in L"?, and second: this is not what the theorem states, but merely "bounded below", which it is. $\endgroup$ – DonAntonio Jun 6 '12 at 2:32
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    $\begingroup$ @DonAntonio: The ordered set in which bounds are considered is $S$ (not $L$, and in the context of the OP's example, not $\mathbb R$). While the statement of the theorem would perhaps be clearer if "in $S$" were explicitly stated for emphasis, in this context $S$ is the "universe". The most common application is where $S=\mathbb R$, but Rudin's statement abstracts the essential order properties, and the OP's example does not satisfy those properties. $\endgroup$ – Jonas Meyer Jun 6 '12 at 2:34
  • $\begingroup$ Got it now! Thanks $\endgroup$ – DonAntonio Jun 6 '12 at 2:37

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