(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.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2F094279f9-3566-4e6f-afb1-7d52184c981e%2F86mjr58_processed.png&w=3840&q=75)
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.)
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