Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.)
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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.)
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