Let Y be a nonempty bounded, open subset of ℝn and let Y¯ denote its closure. Let {Uj}j ≥ 1 be a collection of open sets in ℝn such that Y¯ ⊆ ∪j ≥ 1 Uj. Which of the following statements is always true?

Let α, β ∈ (0, &infty;). Consider the infinite series: ∑n ≥ 4 1n(loge n)α (loge(loge n))β Which of the following statements is true?

For a non-negative real number a, let √a denote its non-negative square root. Consider the sequence {xn}n ≥ 1 defined by x1 = 1 and xn+1 = √(1 + xn2n) for n ≥ 1. Which of the following statements is true?