UCLA Natural Numbers Rational Numbers Subsets Convergence & Limits Worksheet

I’m working on a mathematics multi-part question and need an explanation to help me understand better.

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.