Let A be a nonempty set of real numbers which

In summary, the problem asks to prove that the greatest lower bound of a nonempty set of real numbers A is equal to the negative of the least upper bound of the set -A. The solution involves using the definitions of upper and lower bounds, as well as the least upper and lower bounds, and deriving a contradiction from the assumption that a certain number is an upper bound for -A.
  • #1
Jamin2112
986
12

Homework Statement



Let A be a nonempty set of real numbers which is bounded below. Let -A be the set of all real numbers -x, where x is in A. Prove that inf A = -sup(-A).

Homework Equations



Definitions of upper bound, lower bound, least upper bound, and least lower bound.

The Attempt at a Solution



Here's what I have so far:


Ex5.png



Almost there! I just need to derive a contradiction on my assumption that gamma is an upper bound for -A.
 
Physics news on Phys.org
  • #2
[URL]http://www.threadbombing.com/data/media/8/bummmp.gif[/URL]
By the way, if you help me with this problem I'll give you another one.
 
Last edited by a moderator:

Related to Let A be a nonempty set of real numbers which

1. What does "Let A be a nonempty set of real numbers" mean?

"Let A be a nonempty set of real numbers" is a notation commonly used in mathematics to introduce a set of real numbers, which are the numbers that can be expressed on a number line with both positive and negative values. In this context, "nonempty" means that the set contains at least one element.

2. How is a set of real numbers different from a set of integers?

A set of real numbers includes all numbers that can be expressed on a number line, including fractions and irrational numbers like pi and square roots. A set of integers, on the other hand, only includes whole numbers (positive, negative, and zero).

3. What is the purpose of using "Let A be a nonempty set of real numbers" in a mathematical proof?

Using this notation in a proof allows mathematicians to define a specific set of numbers that is relevant to the problem at hand. This helps to clarify the assumptions and parameters of the problem and allows for a more precise solution.

4. Can A be an infinite set of real numbers?

Yes, A can be an infinite set of real numbers. As long as the set contains at least one real number, it can be considered nonempty. This includes infinite sets such as the set of all positive real numbers or the set of all real numbers between 0 and 1.

5. How does the concept of "nonempty" apply to the set A in this context?

The word "nonempty" is used to ensure that the set A contains at least one element, which is necessary for certain mathematical operations and theorems to hold true. If the set A were empty (containing no elements), it would not be possible to perform these operations or apply these theorems. Therefore, specifying that A is nonempty is important in defining the problem and finding a solution.

Similar threads

Set of least upper bounds multiplied by a constant
    • Calculus and Beyond Homework Help
Replies
1
Views
981
If A is in B then sup(A) < sup(B)
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Supremum, Infimum (Is my proof correct?)
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Show that a set has no "unique nearest point" property
    • Calculus and Beyond Homework Help
Replies
14
Views
690
Proving that there is a sequence in S, such that ##\lim s_n = \sup S##
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
Prove that x is in S or x is an accumulation point of S
    • Calculus and Beyond Homework Help
Replies
25
Views
2K
Can This Proof Verify the Existence of Integers m and n for Any Real Number x?
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
For a set S, there is always a sequence converging to sup(S)
    • Calculus and Beyond Homework Help
Replies
14
Views
2K
Nonempty subset of reals which is bounded
    • Calculus and Beyond Homework Help
Replies
6
Views
2K
Triangle Inequality: use to prove convergence
    • Calculus and Beyond Homework Help
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top