Consider the following functions defined for x y isin r d1x

Question

Math 104: Homework 5-

1. (a) By using the integral test, or otherwise, prove that

_n=2∑^∞1/n(log n)^p

converges if and only if p > 1.

(b) Optional for the enthusiasts. Suppose (an) is a non-increasing sequence of positive real numbers, and that ∑ an converges. By considering the Cauchy criterion, or otherwise, prove that na_n → 0 as n → ∞. By considering part (a), show that the converse result is not true.

2. Consider the following functions defined for x, y ∈ R:

d₁(x, y) = (x - y)², d₂(x, y) = √(|x - y|), d₃(x, y) = |x² - y²|, d₄(x, y) = |x - 2y|.

For each function, determine whether it is a metric or not.

3. Consider two-dimensional space R², with positions written as u = (u₁, u₂), and the Euclidean norm defined as ||u|| = (u₁² + u₂²)^1/2. The Poincare disk model consists of the points S = {u : ||u|| < 1}, with metric

d(u, v) = cosh^-1[1+ (2||u - v||²/(1 - ||u||²)(1 - ||v||²))

for all u, v ∈ S. Define r = cosh^-1 5/4. Draw the Poincare disk, and then calculate and draw the neighborhoods Nr(u) for u = (0, 0), (1/2, 0), and (3/4, 0). [This can be done analytically, although if you prefer, you can also make use of a computer.]

4. Suppose that d₁ and d₂ are equivalent metrics for a set S. Prove that if a sequence (s_n) converges to s with respect to d1, then it also converges with respect to d₂.

5. Suppose that (p_n) is a Cauchy sequence in a set S with metric d, and that some subsequence (p_{n_k}) converges to a point p ∈ S. Prove that the full sequence (p_n) converges to p.

6. Suppose that (p_n) and (q_n) are Cauchy sequences in a set S with metric d. Define (a_n) = d(p_n, q_n). Show that the sequence (an) converges. It may be useful to consider the triangle inequality

d(p_n, q_n) ≤ d(p_n, p_m) + d(p_m, q_m) + d(q_m, q_n)

which is true for all n and m.

7. Optional for the enthusiasts. Consider two-dimensional space R². Define an alternative norm as ||u||_S = (u₁² + u₂² + u₁u₂)^1/2. Prove that the function d_S(u, v) = ||u - v||_S is a metric, and that it is equivalent to the Euclidean metric.