Let S be a nonempty bounded subset of R and let m = sup S. Prove that m is in S iff m = maxS
Let S be a nonempty bounded subset of R and let m = sup S. Prove that m is in S iff m = maxS
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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Section 3.3 Homework
1. Given the following sets, find the maximum, minimum, infimum
(glb) and supremum (lub)of the following sets:
a] [0,4)
b)
d] "inɛN
n+1
el {-15 :nɛN}
(-1)
f]
a feg:r ss}
i feg:0sr and r²ss}
2. Let S be a nonempty bounded subset of R and let m = sup S.
Prove that m is in S iff m = maxS
3. Let S be a nonempty bounded subset of R. Prove that maxS and
supS are unique.
4. Let S be a nonempty bounded subset of R. Let k be a real
number. Define kS = {ks: for some s in S}
Prove the following:
[a] If k >0, then sup(kS) = k(supS) and inf(kS)=k(infS)
[b] if k < 0, then sup(kS) = k(infS) and inf(kS) = k(supS)
5. Let S and T be nonempty bounded subsets of R with S a subset of
T. Prove that infT < infS <supS < supT
6. Prove that between any two real numbers there are infinitely
many rational and irrational numbers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2F2991a9f5-ef18-4ae7-954c-872af41647dc%2Fjvmcxsd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3:50
AA
A 1-xythos.content.blackboardcdn.com
Section 3.3 Homework
1. Given the following sets, find the maximum, minimum, infimum
(glb) and supremum (lub)of the following sets:
a] [0,4)
b)
d] "inɛN
n+1
el {-15 :nɛN}
(-1)
f]
a feg:r ss}
i feg:0sr and r²ss}
2. Let S be a nonempty bounded subset of R and let m = sup S.
Prove that m is in S iff m = maxS
3. Let S be a nonempty bounded subset of R. Prove that maxS and
supS are unique.
4. Let S be a nonempty bounded subset of R. Let k be a real
number. Define kS = {ks: for some s in S}
Prove the following:
[a] If k >0, then sup(kS) = k(supS) and inf(kS)=k(infS)
[b] if k < 0, then sup(kS) = k(infS) and inf(kS) = k(supS)
5. Let S and T be nonempty bounded subsets of R with S a subset of
T. Prove that infT < infS <supS < supT
6. Prove that between any two real numbers there are infinitely
many rational and irrational numbers.
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