Math 447: Real Variables Homework 3

发布时间：2023-12-25

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Homework 3

Math 447: Real Variables

(Exercises were written by Dr. Jason Elliot and are copyrighted by NetMath.)

Exercise 1 Let ϕ : Q+ → R+ (where R+ is a set of Dedekind cuts as deined in the lectures) be the function deined by

ϕ(q) = {a ∈ Q | a < q}.

Note: The two parts of this exercise imply that we can identify a given element q ∈ Q+ with the corresponding Dedekind cut Dq ∈ R+.

(a) Prove that ϕ is injective and order-preserving. That is, prove that if q < q' then ϕ(q) < ϕ(q、).

(b) Prove that ϕ respects both ield operations: ϕ(qr) = ϕ(q)ϕ(r) and ϕ(q + r) = ϕ(q) + ϕ(r).

Exercise 2 Let r 2 R+. For each part, verify your answer using the deintion of convergence.

(a) Calculate lim 1/nr.

(b) Show that lim r = 1.

Exercise 3 Suppose (xn ) is a sequence of real numbers. Consider the following conditions, each of which will either be true or false for any particular sequence:

In some cases two of the above conditions are equivalent. For example, (5) and (6) are equivalent, so (5) ⇐⇒ (6).

In other cases, there is a strictly weaker/strictly stronger relation between them. For example, (1) =⇒ (3), but not conversely (that is, (3)

For each pair of statements, determine whether one implies the other and vice versa. For example, determine =⇒ (1)). whether (4) =⇒ (7) and vice versa whether (7) =⇒ (4). Do this for all pairs of statements. Give a valid proof of each comparison, including a proof that (5) ⇐⇒ (6) and that (1) =⇒ (3), but not conversely.

Tips: 1. You can make your work easier by using transitivity of “ =⇒”).

2. A good way to prove that (3) =⇒ (1) is to find a sequence that satisfies (3) but violates (1). Likewise for any other similar relation.