Which of the following is an inner product on the vector space of all real-valued continuous functions on [0, 1]?

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?

Let (ℝ, d) be the usual metric space. Which of the following subsets of ℝ is NOT compact?

Let &mathbb;N denote the set of all positive integers. For n ∈ &mathbb;N, let fn : [0, 1] → &mathbb;R be given by fn(x) = n(1 - nx) if 0 ≤ x ≤ 1/n, and 0 if x > 1/n. Which of the following statements is necessarily true?

For each positive integer n, let fn: [-1, 1] → &mathbb;R be given by fn(x) = -1 if -1 ≤ x ≤ -1/n, fn(x) = nx if -1/n < x < 1/n, and fn(x) = 1 if 1/n ≤ x ≤ 1. On which of the following intervals does the sequence {fn}n ≥ 1 converge uniformly?

Which of the following statements is necessarily true?

Let S be a subset of the open interval (0, 1) consisting of all real numbers α = 0.a1a2a3... whose infinite decimal expansion satisfies ai ∈ {0, 2, 4} for all i ≥ 1. Which of the following statements is true?

Let f : (0, ∞) → (0, ∞) be a uniformly continuous function. Define g : (0, ∞) → (0, ∞) by g(x) = x2 if 0 1. Which of the following statements is true?