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If $A, B$ are two non empty sets of real numbers and for every $a$ from $A$ and $b$ from $B$

$a < b$,

how can I prove that

$\sup A \leq \inf B$?

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This is not true; you can only say that $supA \le infB$. –  Gadi A Sep 13 '10 at 18:21
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This question is part of a series that sound like homework problems. –  whuber Sep 13 '10 at 18:22
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Looking at your history this is a homework problem! –  alext87 Sep 13 '10 at 18:24
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-1. Do your own homework, or ask conceptual questions. –  Qiaochu Yuan Sep 13 '10 at 20:57
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4 Answers 4

up vote 6 down vote accepted

I don't think this is true. Let $A=(0,1)$ and $B=(1,2)$ then $supA=1$ and $infB=1$ but $A$ and $B$ satisfy your conditions.

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The conditions are violated because 1 is in A and is not strictly less than all elements of B. –  whuber Sep 13 '10 at 18:48
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@whuber 1 is in A=(0,1)? Shurely shome mishtake? –  Bob Durrant Sep 13 '10 at 19:27
    
@Bob: Thank you: I was misreading these as doubleton sets, not intervals. Alext87's answer is a good one. –  whuber Sep 13 '10 at 20:37
    
@alext87, this question is absolutely correct. In A=(0,1) and B=(1,2), supA=infB. The question wants us to prove that supA<=infB. (It includes '=' case). –  Nipun Malhotra Sep 11 '11 at 13:47
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@Nipun: The question was edited after alext87's answer was given (and presumably accepted), as shows the edit history of the question. –  Asaf Karagila Sep 11 '11 at 14:54
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With the edited problem: if $a\lt b$ for all $a\in A$ and $b\in B$, why is it that $\sup(A)\leq\inf(B)$? Is every element of $A$ a lower bound for $B$? If so, what does that tell you about the elements of $A$ relative to $\inf(B)$? And given that, what does that tell you about $\sup(A)$?

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Here's the solution:

Let us suppose that $\sup(A) > \inf(B)$. So, there exists some $x∈A$, such that $\inf(B) < x$. Again, as $\inf(B)< x$, there exists $y∈B$, such that $y<x$. which is a contradiction as we have been given that $A<B$ (i.e, $x<y$ for all $x∈A$, $y∈B$). So, our assuption that $\sup(A) > \inf(B)$ is WRONG. So, $\sup(A) \leq \inf(B)$. Hence Proved...... :)

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I have converted the first part of your answer into a comment to alext87 (look above). Because you do not have 50 reputation points yet, you can only comment on your own questions and answers. So, you didn't do anything wrong; the "add comment" button will only appear for you once you gain 50 points. Here is an explanation of reputation points. –  Zev Chonoles Sep 11 '11 at 15:15
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Consider $b\in B$, by hypothesis is a upper bound of $A$, so by definition of sup (the least upper bound), $\sup A\leq b,$ $ \forall b\in B$. Now the $\sup A$ is a lower bound of $B$, so by defnition of inf(the grater lower bound), $\sup A\leq \inf B$ .

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