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
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
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2Fc75f061a-9466-489b-b09d-3e8994bf27a3%2F21duk1n_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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