Problem 1. A set ICR is called an interval if for all triples of real numbers x, y, and z such that x < z < y, if x € I and y € I then z € I. Prove that every interval takes exactly one of the following nine forms: (-∞,b), (-∞,b], (a,b), (a,b], [a,b), [a,b], (a, ∞), [a, ∞), or (-∞, ∞).
Problem 1. A set ICR is called an interval if for all triples of real numbers x, y, and z such that x < z < y, if x € I and y € I then z € I. Prove that every interval takes exactly one of the following nine forms: (-∞,b), (-∞,b], (a,b), (a,b], [a,b), [a,b], (a, ∞), [a, ∞), or (-∞, ∞).
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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![Problem 1. A set I ≤ R is called an interval if for all triples of real numbers x, y,
and z such that x < z < y, if x € I and y € I then z € I. Prove that every interval takes
exactly one of the following nine forms:
(-∞,b), (-∞,b], (a,b), (a,b], [a,b), [a,b], (a, ∞), [a, ∞), or (-∞, ∞).
(Hint: You have to consider an arbitrary set I satisfying the condition in the problem and
prove something about it. Consider cases according to whether I is (unbounded above /
bounded above and contains its sup / bounded above and does not contain its sup), and
similarly for below. You should have 3-3 = 9 cases.")](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F572394d6-763b-4634-a070-d7c873f29a55%2Fc66070b8-c819-4915-988d-4e373a5ce3bc%2Fp4u210u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1. A set I ≤ R is called an interval if for all triples of real numbers x, y,
and z such that x < z < y, if x € I and y € I then z € I. Prove that every interval takes
exactly one of the following nine forms:
(-∞,b), (-∞,b], (a,b), (a,b], [a,b), [a,b], (a, ∞), [a, ∞), or (-∞, ∞).
(Hint: You have to consider an arbitrary set I satisfying the condition in the problem and
prove something about it. Consider cases according to whether I is (unbounded above /
bounded above and contains its sup / bounded above and does not contain its sup), and
similarly for below. You should have 3-3 = 9 cases.")
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