Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(a) Prove that [a, b] is sequentially compact, but (a, b) is not sequentially compact. (Hint: For [a, b], Bolzano-Weierstrass might be helpful. For (a, b), try constructing a convergent sequence which 'escapes' the open interval.)

(a) Prove that [a, b] is sequentially compact, but (a, b) is not sequentially compact. (Hint: For [a, b], Bolzano-Weierstrass might be helpful. For (a, b), try constructing a convergent sequence which 'escapes' the open interval.)

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 72E
See similar textbooks
icon
Related questions
Question
In this problem, we will see how to generalize some of the results in section 3.3 to more general domains. We say a set KCR is sequentially compact if every sequence in K has a convergent subsequence converging to a point in K. In other words, if {n} is a sequence satisfying x₁ € K for all n € N, then there exists a subsequence {n} and point x € K such that → as k→ ∞. (a) Prove that [a, b] is sequentially compact, but (a, b) is not sequentially compact. (Hint: For [a, b], Bolzano-Weierstrass might be helpful. For (a, b), try constructing a convergent sequence which 'escapes' the open interval.) (b) Prove that if a set K is sequentially compact, then it is bounded. (Hint: The contrapositive might be easier to prove, look at the proof of Lemma 3.3.1.) (c) Prove that if a set K is sequentially compact, then sup(K) € K and inf(K) € K. (Hint: Try constructing sequences which converge to sup(K) and inf(K).) (d) Let K be sequentially compact. Prove that if a function f : K → R is continuous, then the direct image f(K) is sequentially compact. Use this to prove that f achieves an absolute minimum and absolute maximum on K. (Hint: For any sequence {f(n)}, observe that {n} is a sequence in the sequen- tially compact set K.)
Transcribed Image Text:In this problem, we will see how to generalize some of the results in section 3.3 to more general domains. We say a set KCR is sequentially compact if every sequence in K has a convergent subsequence converging to a point in K. In other words, if {n} is a sequence satisfying x₁ € K for all n € N, then there exists a subsequence {n} and point x € K such that → as k→ ∞. (a) Prove that [a, b] is sequentially compact, but (a, b) is not sequentially compact. (Hint: For [a, b], Bolzano-Weierstrass might be helpful. For (a, b), try constructing a convergent sequence which 'escapes' the open interval.) (b) Prove that if a set K is sequentially compact, then it is bounded. (Hint: The contrapositive might be easier to prove, look at the proof of Lemma 3.3.1.) (c) Prove that if a set K is sequentially compact, then sup(K) € K and inf(K) € K. (Hint: Try constructing sequences which converge to sup(K) and inf(K).) (d) Let K be sequentially compact. Prove that if a function f : K → R is continuous, then the direct image f(K) is sequentially compact. Use this to prove that f achieves an absolute minimum and absolute maximum on K. (Hint: For any sequence {f(n)}, observe that {n} is a sequence in the sequen- tially compact set K.)
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS