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.

Let $A$ be any non-empty subset of $\mathbb{R}$. Then $s = \sup A$ iff $s$ has the following properties:

  1. $s \geq a$ for every $a \in A$,

  2. if $t < s$, then there exists an $a \in A$ such that $a > t$.

Prove it? having problem in proving 2.

share|improve this question
    
What does "If tt" mean? –  joriki Oct 1 '12 at 10:11
    
Huh? What is your condition (2) meant to say, and what is your definition of 'sup A'? –  Billy Oct 1 '12 at 10:12
    
@robjohn: I'm pinging you for Brian's comment to Michael (since you were the one approving it). –  Asaf Karagila Oct 1 '12 at 10:43
    
@BrianM.Scott: I didn't make any conscious changes to the mathematical content, I only Latexed the symbols and put (1) and (2) into a list. There was certainly a lot more than 'If tt' written for (2) when I edited the question. –  Michael Albanese Oct 1 '12 at 10:47
1  
@Michael: You’re right. Very weird. Anyhow, I’ve restored your edit. Sorry about all the confusion. –  Brian M. Scott Oct 1 '12 at 11:02

3 Answers 3

Hint: If $t<s$, then prove that there exists $h>0$(however small), such that $(t+h)<s$, and this $(t+h)$ will be your required $a$.You can refer Analysis by Terence Tao for detailed explanation.

share|improve this answer

Step 1: write down the definition of $\sup$ of a set.

Step 2: Prove the $\implies$ direction. To do this, assume $s = \sup A$. Then show that $s$ satisfies 1. and 2.

Step 3: Prove the $\Longleftarrow$ direction. To do this, assume that $s$ satisfies 1. and 2. and show that $s = \sup A$.

Once you have done that you have solved the question, that's all you need to do.

share|improve this answer

Whatever it is, compare it to the definition given for the $\sup$, try to deduce one from the other.

share|improve this answer

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.