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THEOREM 11.1.2 If M is the least upper bound of the set S and e is a positive number, then there is at least one number s in S such that M-e< s < M. AXIOM 11.1.1 THE LEAST UPPER BOUND AXIOM Every nonempty set of real numbers that has an upper bound has a least upper bound. THEOREM 11.3.4 Every convergent sequence is bounded. THEOREM 11.3.6 A nondecreasing sequence which is bounded above converges to the least upper bound of its range. A nonincreasing sequence which is bounded below converges to the greatest lower bound of its range.

THEOREM 11.1.2 If M is the least upper bound of the set S and e is a positive number, then there is at least one number s in S such that M-e< s < M. AXIOM 11.1.1 THE LEAST UPPER BOUND AXIOM Every nonempty set of real numbers that has an upper bound has a least upper bound. THEOREM 11.3.4 Every convergent sequence is bounded. THEOREM 11.3.6 A nondecreasing sequence which is bounded above converges to the least upper bound of its range. A nonincreasing sequence which is bounded below converges to the greatest lower bound of its range.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 28E
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explain of logical relationships of (11.1.1), (11.1.2), (11.3.4), (11.3.6)

THEOREM 11.1.2 If M is the least upper bound of the set S and e is a positive number, then there is at least one number s in S such that M-e< s < M. AXIOM 11.1.1 THE LEAST UPPER BOUND AXIOM Every nonempty set of real numbers that has an upper bound has a least upper bound. THEOREM 11.3.4 Every convergent sequence is bounded. THEOREM 11.3.6 A nondecreasing sequence which is bounded above converges to the least upper bound of its range. A nonincreasing sequence which is bounded below converges to the greatest lower bound of its range.
Transcribed Image Text:THEOREM 11.1.2 If M is the least upper bound of the set S and e is a positive number, then there is at least one number s in S such that M-e< s < M. AXIOM 11.1.1 THE LEAST UPPER BOUND AXIOM Every nonempty set of real numbers that has an upper bound has a least upper bound. THEOREM 11.3.4 Every convergent sequence is bounded. THEOREM 11.3.6 A nondecreasing sequence which is bounded above converges to the least upper bound of its range. A nonincreasing sequence which is bounded below converges to the greatest lower bound of its range.
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