Upper bounds of a non-empty set
Not what you're looking for?
Let S be a non-empty set of real numbers, and prove that the following statements are equivalent:
(1) If v is any upper bound of S, then u <= v (read as "u is less than or equal to v").
(2) If z < u, then z is not an upper bound of S.
(3) If z < u, then there exists s_z (read as "s sub z") in S such that z < s_z.
(4) If epsilon > 0, then there exists s_epsilon (read as "s sub epsilon") in S such that u - epsilon < s_epsilon.
Purchase this Solution
Solution Summary
A detailed proof of the equivalence of the four given statements is provided. Specifically, the proof consists of showing the following: (a) statement (1) implies statement (2); (b) statement (2) implies statement (3); (c) statement (3) implies statement (4); (d) statement (4) implies statement (1).
Solution Preview
To prove that statements (1) through (4) are equivalent, we will prove the following:
(1) -> (2)
(2) -> (3)
(3) -> (4)
(4) -> (1)
--------------------------------------------------------------
Proof that (1) -> (2):
Assume that (1) holds, and prove that (2) holds.
Suppose (by way of contradiction) that (2) does not hold. Then there exists z < u such that z is an upper bound of S. Setting v to z in (1), we obtain u <= z, since z is an upper bound of S, but then we have u <= z < u, which is absurd, since the statements "u <= z" and "u > z" cannot both be true. Thus our supposition that (2) does not hold is false, so (2) ...
Education
- AB, Hood College
- PhD, The Catholic University of America
- PhD, The University of Maryland at College Park
Recent Feedback
- "Thanks for your assistance. "
- "Thank you. I understand now."
- "Super - Thank You"
- "Very clear. I appreciate your help. Thank you."
- "Great. thank you so much!"
Purchase this Solution
Free BrainMass Quizzes
Basic Social Media Concepts
The quiz will test your knowledge on basic social media concepts.
Paradigms and Frameworks of Management Research
This quiz evaluates your understanding of the paradigm-based and epistimological frameworks of research. It is intended for advanced students.
Situational Leadership
This quiz will help you better understand Situational Leadership and its theories.
Transformational Leadership
This quiz covers the topic of transformational leadership. Specifically, this quiz covers the theories proposed by James MacGregor Burns and Bernard Bass. Students familiar with transformational leadership should easily be able to answer the questions detailed below.
Business Ethics Awareness Strategy
This quiz is designed to assess your current ability for determining the characteristics of ethical behavior. It is essential that leaders, managers, and employees are able to distinguish between positive and negative ethical behavior. The quicker you assess a person's ethical tendency, the awareness empowers you to develop a strategy on how to interact with them.