Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Prove that for any nonempty set $S \subset \mathbb{R}$, $ \inf(S)\leq \sup(S)$ and give necessary and sufficient conditions for equality.

This is what I have so far but I think I am on the wrong track:

Since set S is contained in R, we have four options: $$S=(a,b) ; S=[a,b) ; S=(a,b] ; S=[a,b]$$ for some $ a \text{ and } b \in \mathbb{R}$

via the ordering of interval notation $\inf(S)=a$ and $\sup(S)=b$ and $a\leq b$ by definition of interval notation. Hence $\inf(S) \leq \sup(S)$.

share|improve this question
1  
Question about sets? Definitely [set-theory]... –  Asaf Karagila Jan 21 '13 at 0:23
6  
$S$ need not be an interval. –  Shaun Ault Jan 21 '13 at 0:24
4  
Surely you mean that $\inf S\le\sup S$. –  Brian M. Scott Jan 21 '13 at 0:29
1  
@Asaf: Nah, it must be [tennis]. –  Brian M. Scott Jan 21 '13 at 0:31
8  
The vote to close this as not a real question is incomprehensible. Not only is the question perfectly clear (apart from the unfortunate typo), but the OP has provided an attempt at a solution. –  Brian M. Scott Jan 21 '13 at 0:36

2 Answers 2

up vote 6 down vote accepted

I'm assuming you want to show that $\inf S\leq \sup S$, whenever $S\subseteq \mathbb{R}$ is a non-empty subset, and not the reverse inequality.

A part of the definition of the supremum is that it is an upper bound for $S$, i.e. $$s\leq \sup S\quad \text{for all }\;s\in S.$$ The same goes for the infimum, it is a lower bound for $S$: $$\inf S\leq s\quad\text{for all }\; s\in S.$$

Now use the fact that $S$ is non-empty to deduce the inequality.

share|improve this answer
2  
I love when math is written this obviously. Nice job, Stefan. –  000 Jan 21 '13 at 0:51
1  
Thanks for your response :) –  math101 Jan 21 '13 at 0:51

Hint: Suppose that $S$ has more than one point. If $a,b \in S$ and $ a<b$ what can you deduce about the $\inf$ and $\sup$ of $S$? What happens if $S$ has one point?

share|improve this answer
    
Thanks Jacob :) –  math101 Jan 21 '13 at 0:53

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.