Let X = &mathbb{R} with the usual metric, and for a ∈ X and r > 0, let Br(a) = {x ∈ X : d(x, a) 0 such that Br(x) ∩ Br(y) = φ. Q: If E is a closed subset of &mathbb{R}$, then there exists r > 0 such that E ⊆ Br(0). Which of the statements is/are 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?