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.
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