MATH 242 – Principles of Analysis Problem Set 3 – due: Feb. 14

‘A’ Section

1. A set B is said to be finite if there is some n ∈ N (the number of elements in B), and a one-to-one and onto mapping f : {1, 2, …, n} → B. (Intuitively, we think that f(1) = b_{1}, f(2) = b_{2}, … “counts through” all the elements of B one at a time without repetitions and without missing any elements in B.) For each of the following sets, either show B is finite by determining the n and constructing a mapping f as above, or say why no such mapping exists. a. B = {r = p/q ∈ Q | 1 ≤ q ≤ 4 and 0 < r < 1} b. B = {r = p/q ∈ Q | 0 < r < 1} c. B = {n ∈ Z | |n| ≤ 10^{14}}

2. Which of the following sequences converge to 0? Explain your answers, but you do not need to provide complete formal proofs of your assertions. a. {x_{n}}, where

x_{n} = \{ \begin{matrix}2^{n} & if n ≤ 1000 \\ 2^{−n} & if n > 1000 \end{matrix}

b. {y_{n}}, where

y_{n} = \{ \begin{matrix}1 & if n is evenly divisible by 100 \\ \frac{1}{n} & if n is not evenly divisible by 100 \end{matrix}

c. {z_{n}}, where

z_{n} = \{ \begin{matrix}n & if n is a Mersenne prime number \\ \frac{(−1)^{n}}{n^{2}} & if n is not a Mersenne prime number \end{matrix}

(look these up on Wikipedia and read about them.)

3. Let f(x) = [x] be the greatest integer function, defined as [x] = the greatest integer ≤ x. a. If x_{n} → a, does it follow that [x_{n}] → [a]? Prove or give a counterexample. b. If [x_{n}] → [a], does it follow that x_{n} → a? Prove or give a counterexample.

(over for ‘B’ Section problems)

‘B’ Section

1.

a. Prove that \sqrt{3} is an irrational number.

b. If r ≠ 0 and s are rational numbers, show that r\sqrt{3} + s is also an irrational

number. (Hint: Suppose not and derive a contradiction.) c. If x = r\sqrt{3} + s and x′ = r′\sqrt{3} + s′ are two numbers as in part b, what can be

said about x + x′ and xx′? Are they necessarily irrational too?

2. Let A and B be two nonempty sets of real numbers.

a. Assume that x ≤ y for all x ∈ A and y ∈ B. Show that lub A and glb B must

exist. b. Under the same assumptions as part a, show that lub A ≤ glb B. c. Now assume that A and B are bounded. Is it true that lub A ≤ glb B implies

that x ≤ y for all x ∈ A and y ∈ B? Prove or give a counterexample.

3. Let A be a bounded set of real numbers and let B = {kx | x ∈ A}, where k < 0 is a strictly negative number. Show that B is also bounded. Then, determine formulas for computing lub B and glb B in terms of lub A and glb A, and prove your assertions.

4. Determine whether each of the following sequences converge and prove your assertions using the ε, n_{0} definition of convergence. a. x_{n} = \frac{3n^{2}}{n^{2}+5} b. x_{n} = \frac{1}{ln(n)} c. x_{n} = cos(nπ).