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## Natural Numbers Rational Numbers Subsets Convergence

Order ID# 45178248544XXTG457Plagiarism Level: 0-0.5%Natural Numbers Rational Numbers Subsets ConvergenceWriter Classification: PhD competentStyle: APA/MLA/Harvard/ChicagoDelivery: Minimum 3 HoursRevision: PermittedSources: 4-6Course Level: Masters/University CollegeGuarantee Status: 96-99%

Instructions1. Prove that 2^n > n^3 for all natural numbers n ? 10.

2. Prove that sqrt(11 ? sqrt13) is not a rational number.

3.Let A, B ? R be non-empty sets. We define AB = {ab | a ? A, b ? B} .Assume that both A,B are bounded and subsets of [0,?) = {x ? R | x ? 0}.

a.Prove that AB is bounded.

b.Prove that sup(AB) = sup(A) sup(B), inf(AB) = inf(A) inf(B). You can use part a, even if you did not solve it.

4. Prove that lim_n?? (?n3 +2n2 ?3n+4) / (6n3 +2n2 ?2n+3) =?1 /6 using the definition of convergence without using limit theorems.

5. For each of the following, give an example of a pair of sequences (an)n?N,(bn)n?N with the prescribed property. You should show that your example indeed satisfies the property.

a. lim_n?? an = +?, lim_n?? bn = 0, lim_n?? anbn < 0.

b.Both (an)_n?N, (bn)_n?N diverge, but (anbn)_n?N converges.

c. We have lim_ n?? an =2, lim_ n??bn =3, and moreover, an <2, bn >3, anbn >6 for all n ? N.

RUBRIC

Excellent Quality95-100%

Introduction45-41 points

The background and significance of the problem and a clear statement of the research purpose is provided. The search history is mentioned.

Literature Support91-84 points

The background and significance of the problem and a clear statement of the research purpose is provided. The search history is mentioned.

Methodology58-53 points

Content is well-organized with headings for each slide and bulleted lists to group related material as needed. Use of font, color, graphics, effects, etc. to enhance readability and presentation content is excellent. Length requirements of 10 slides/pages or less is met.

Average Score50-85%

40-38 points

More depth/detail for the background and significance is needed, or the research detail is not clear. No search history information is provided.

83-76 points

Review of relevant theoretical literature is evident, but there is little integration of studies into concepts related to problem. Review is partially focused and organized. Supporting and opposing research are included. Summary of information presented is included. Conclusion may not contain a biblical integration.

52-49 points

Content is somewhat organized, but no structure is apparent. The use of font, color, graphics, effects, etc. is occasionally detracting to the presentation content. Length requirements may not be met.

Poor Quality0-45%

37-1 points

The background and/or significance are missing. No search history information is provided.

75-1 points

Review of relevant theoretical literature is evident, but there is no integration of studies into concepts related to problem. Review is partially focused and organized. Supporting and opposing research are not included in the summary of information presented. Conclusion does not contain a biblical integration.

48-1 points

There is no clear or logical organizational structure. No logical sequence is apparent. The use of font, color, graphics, effects etc. is often detracting to the presentation content. Length requirements may not be met

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Natural Numbers Rational Numbers Subsets Convergence

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