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Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.

Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Lower bound. Infimum. Carefully read the definitions and remarks. Then attempt Question 2 at the end of the sub-section. Definition 0.1. Lower bound: Given a subset S C R. A real number a eR is a lower bound for the set ScR if a < s for each s e S. Definition 0.2. Infimum: Given a subset S C R. The real number y E R is the lower bound of the set SCRif (1) y < s for each s e S (i.e. y is a lower bound for S), and (2) if a is any lower bound for S, then y > a. Remark 1. We write y = inf S S if the real number y is infimum or the greatest lower bound of a subset S C R. The Latin word infimum is used to denote the same quantity, abbreviated to inf: y = inf S. The subset s CR is called bounded from below if it has a lower bound. Remark 2. It follows from the Well-Ordering Principle that every set of real numbers that is nonempty and bounded from below has an infimum: As a matter of fact, a subset S C R is bounded from below iff the set S' := {x: -x € S} is bounded from above, and if S is nonempty and bounded from below, then – inf S' is the infimum of S. Remark 3. If the subset S c R has a smallest element, then inf S is simply the smallest element of S, often denoted min S. (For instance, if S is nonempty and finite.)
Transcribed Image Text:Lower bound. Infimum. Carefully read the definitions and remarks. Then attempt Question 2 at the end of the sub-section. Definition 0.1. Lower bound: Given a subset S C R. A real number a eR is a lower bound for the set ScR if a < s for each s e S. Definition 0.2. Infimum: Given a subset S C R. The real number y E R is the lower bound of the set SCRif (1) y < s for each s e S (i.e. y is a lower bound for S), and (2) if a is any lower bound for S, then y > a. Remark 1. We write y = inf S S if the real number y is infimum or the greatest lower bound of a subset S C R. The Latin word infimum is used to denote the same quantity, abbreviated to inf: y = inf S. The subset s CR is called bounded from below if it has a lower bound. Remark 2. It follows from the Well-Ordering Principle that every set of real numbers that is nonempty and bounded from below has an infimum: As a matter of fact, a subset S C R is bounded from below iff the set S' := {x: -x € S} is bounded from above, and if S is nonempty and bounded from below, then – inf S' is the infimum of S. Remark 3. If the subset S c R has a smallest element, then inf S is simply the smallest element of S, often denoted min S. (For instance, if S is nonempty and finite.)
Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.
Transcribed Image Text:Question 2. Prove that a nonempty closed subset of R, if it is bounded from below, has a least element.
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