Natural Numbers Rational Numbers Subsets Convergence
Order ID: 89JHGSJE83839 Style: APA/MLA/Harvard/Chicago Pages: 5-10 Instructions:
Natural Numbers Rational Numbers Subsets Convergence
1. 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.Natural Numbers Rational Numbers Subsets Convergence
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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