Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

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
See similar textbooks
icon
Related questions
Question
Show that the subset of R2 given by {(x, y) E R² : xy= 1, x > 0} is closed. 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 SC R. A real number a E R is a lower bound for the set S CR if as 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 1 (2) if a is any lower bound for S, then y 2 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 CR 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:Show that the subset of R2 given by {(x, y) E R² : xy= 1, x > 0} is closed. 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 SC R. A real number a E R is a lower bound for the set S CR if as 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 1 (2) if a is any lower bound for S, then y 2 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 CR 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.)
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS