/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Let \(S\) be a nonempty bounded ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

Let \(S\) be a nonempty bounded subset of \(\mathbb{R}\). (a) Prove that inf \(S \leq\) sup \(S\). Hint: This is almost obvious; your proof should be short. (b) What can you say about \(S\) if inf \(S=\) sup \(S ?\)

Short Answer

Expert verified
inf 饾悞 鈮� sup 饾悞. If inf 饾悞 = sup 饾悞, then 饾悞 is a singleton set.

Step by step solution

01

Understanding Definitions

Recall that for a nonempty bounded subset 饾悞 of \( \mathbb{R} \), sup 饾悞 is the least upper bound and inf 饾悞 is the greatest lower bound of 饾悞. This means sup 饾悞 is the smallest number that is greater than or equal to every element in 饾悞, and inf 饾悞 is the largest number that is less than or equal to every element in 饾悞.
02

Prove inf 饾悞 鈮� sup 饾悞

Since 饾悞 is a subset of \( \mathbb{R} \), there exists at least one element s in 饾悞. By definition of infimum, inf 饾悞 is less than or equal to every element in 饾悞. Similarly, by definition of supremum, sup 饾悞 is greater than or equal to every element in 饾悞. Consequently, inf 饾悞 must be less than or equal to sup 饾悞 because there cannot be any element in 饾悞 that defies this relationship.
03

Analyze inf 饾悞 = sup 饾悞

If inf 饾悞 equals sup 饾悞, then every element in 饾悞 must be exactly the same number, since inf 饾悞 and sup 饾悞 both bound the elements of 饾悞 from below and above, respectively. Therefore, 饾悞 must be a singleton set, meaning it contains exactly one unique element.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

bounded sets
In mathematics, a set is considered 'bounded' if there are real numbers that serve as the limits of the set. Specifically, a set S of real numbers is bounded if there exists an upper bound and a lower bound.

Upper bounds and lower bounds are crucial concepts in understanding bounded sets:
  • An upper bound of S is a number that is greater than or equal to every number in S.
  • A lower bound of S is a number that is less than or equal to every number in S.
For example, if we have the set S = {1, 2, 3}, this set is bounded because it has:
  • a lower bound of 1 or any number less than 1, and
  • an upper bound of 3 or any number greater than 3.
One of the most common examples of a bounded set is a closed interval \([a, b]\) in the real numbers, where all the values are between two finite numbers a and b.
supremum
The supremum (often abbreviated as 'sup') of a set is the least upper bound of that set.

Formally, if you have a set S, then sup S is a number such that:
  • sup S is an upper bound of S, and
  • no number less than sup S is an upper bound of S.
In simpler terms, sup S is the smallest number that is still greater than or equal to every element in S.

For example, if S = {1, 2, 3}, then sup S = 3 because 3 is the smallest number that is greater than or equal to all elements in the set.

If S is a bounded subset of the real numbers, the supremum of S will always exist and will also be a real number.
infimum
The infimum (often abbreviated as 'inf') of a set is the greatest lower bound of that set.

Let鈥檚 consider a set S. The inf S is a number such that:
  • inf S is a lower bound of S, and
  • no number greater than inf S is a lower bound of S.
In essence, inf S is the largest number that is still less than or equal to every element in S.

For instance, if S = {1, 2, 3}, then inf S = 1 because 1 is the largest number that is less than or equal to all elements in the set.

Similar to the supremum, if S is a bounded subset of the real numbers, the infimum of S will always exist and will be an actual real number.
real numbers
Real numbers, denoted as \(\mathbb{R}\), include all the numbers that can be found on the number line. This includes both rational numbers (like 1/2, 4, -7) and irrational numbers (like 鈭�2, 蟺).

Properties of real numbers include:
  • Closed under addition and multiplication: Adding or multiplying two real numbers results in a real number.
  • Order Property: Real numbers are ordered, meaning for any two real numbers, one is either less than, equal to, or greater than the other.
  • Density: Between any two real numbers, there exists another real number.
When solving calculus problems, knowing these properties helps in proving concepts about bounded sets, supremum, and infimum because all these concepts rely on the behavior of real numbers.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For each set below that is bounded above, list three upper bounds for the set. Otherwise write "NOT BOUNDED ABOVE" Or "NBA." (a) [0,1] (b) (0,1) (c) \\{2,7\\} (d) \(\\{\pi, e\\}\) (e) \(\left\\{\frac{1}{n}: n \in \mathbb{N}\right\\}\) (f) \\{0\\} (g) [0,1]\(\cup[2,3]\) (h) \(\cup_{n=1}^{\infty}[2 n, 2 n+1]\) (i) \(\cap_{n=1}^{\infty}\left[-\frac{1}{n}, 1+\frac{1}{n}\right]\) (j) \(\left\\{1-\frac{1}{3^{n}}: n \in \mathbb{N}\right\\}\) (k) \(\left\\{n+\frac{(-1)^{n}}{n}: n \in \mathbb{N}\right\\}\) (l) \(\\{r \in \mathbb{Q}: r<2\\}\) (m) \(\left\\{r \in \mathbb{Q}: r^{2}<4\right\\}\) (n) \(\left\\{r \in \mathbb{Q}: r^{2}<2\right\\}\) (o) \(\\{x \in \mathbb{R}: x<0\\}\) (p) \(\left\\{1, \frac{\pi}{3}, \pi^{2}, 10\right\\}\) (q) \(\\{0,1,2,4,8,16\\}\) (r) \(\cap_{n=1}^{\infty}\left(1-\frac{1}{n}, 1+\frac{1}{n}\right)\) (s) \(\left\\{\frac{1}{n}: n \in \mathbb{N} \text { and } n \text { is prime }\right\\}\) (t) \(\left\\{x \in \mathbb{R}: x^{3}<8\right\\}\) (u) \(\left\\{x^{2}: x \in \mathbb{R}\right\\}\) (v) \(\left\\{\cos \left(\frac{n \pi}{3}\right): n \in \mathbb{N}\right\\}\) (w) \(\left\\{\sin \left(\frac{n \pi}{3}\right): n \in \mathbb{N}\right\\}\)

Prove \(1^{2}+2^{2}+\dots+n^{2}=\frac{1}{6} n(n+1)(2 n+1)\) for all natural numbers \(n\)

Show that if \(\alpha\) and \(\beta\) are Dedekind cuts, then so is \(\alpha+\beta=\left\\{r_{1}+r_{2}:\right.\) \(\left.r_{1} \in \alpha \text { and } r_{2} \in \beta\right\\}\)

Prove that if \(a>0,\) then there exists \(n \in \mathbb{N}\) such that \(\frac{1}{n}

Let \(\alpha=0^{*} \cup\left\\{p \in \mathbb{Q}: p \geq 0 \text { and } p^{2}<2\right\\} .\) Prove that \(\alpha\) is a Dedekind cut and also that it has the property \(\alpha \cdot \alpha=2^{*} ;\) that is, the square of \(\alpha\) is \(2^{*} .\) Note: This seems to be surprisingly tricky, as pointed out by Linda Hill and Robert J. Fisher at Idaho State University. Their solution is available from them or from the author.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.