Question 5 Real AnalysisProof 3. Let CCR be a non-empty bounded subset of integers (where boundedmeans C is bounded above and bounded below). Prove that sup C is aninteger.4. Let D= {d,}nEN be a set of real numbers satisfying dn+1 2 dn +1 for allne N. Prove that D is an unbounded subset of R.5. Let E= {en}n€N be a set of positive real numbers satisfying en+1for all n e N. Prove that inf E = 0.en%3Den+1

Name: Question 5 Real AnalysisProof 3. Let CCR be a non-empty bounded subset
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Description: Question 5 Real AnalysisProof 3. Let CCR be a non-empty bounded subset of integers (where boundedmeans C is bounded above and bounded below). Prove that sup C is aninteger.4. Let D= {d,}nEN be a set of real numbers satisfying dn+1 2 dn +1 for allne N

Answered: 5. Let E {en}n€N be a set of positive…

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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Question 5 Real Analysis Proof

3. Let CCR be a non-empty bounded subset of integers (where bounded
means C is bounded above and bounded below). Prove that sup C is an
integer.
4. Let D= {d,}nEN be a set of real numbers satisfying dn+1 2 dn +1 for all
ne N. Prove that D is an unbounded subset of R.
5. Let E= {en}n€N be a set of positive real numbers satisfying en+1
for all n e N. Prove that inf E = 0.
en
%3D
en+1

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I will write 2 proofs.

One which uses limit theorems and other by explicitly solving the recursive relation.

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