MATH 242 – Principles of Analysis Problem Set 2 – due: Feb. 8

‘A‘ Section

1. Let x ∈ [1, 3]. Determine the largest and smallest values of |x − 5|, |x + 5|, and 1/|x^{2} − 25|.

2. Use the binomial theorem (Theorem 1.4.1) for all parts of this problem. a. Expand using the binomial theorem and simplify as much as possible:

(a^{2} − 5b^{3})^{6}.

b. What is the coefficient of x^{3} in the expansion of

(\frac{x^{4} + 7x}{x^{2}})^{3} .

c. What is ∑_{k=0}^{n} \binom{n}{k}? Explain. d. What is ∑_{k=0}^{n}(−1)^{k}\binom{n}{k}? Explain.

3. For each of the following statements, say whether the statement is true or false. If it is false, give a counterexample; if it is true, give a short reason. a. A set A ⊂ R is bounded if there exists some B such that x ≤ B for all x ∈ A. b. If A, B ⊂ R are bounded, then A ∪ B is also bounded. c. If A, B ⊂ R are bounded, then D = {x − y | x ∈ A, y ∈ B} is also bounded. d. If A, B ⊂ R_{>0} are bounded, then Q = {x/y | x ∈ A, y ∈ B} is also bounded.

4.

a. Let A = [0, 4] ∩ (1, 5). What is a = lub A? What is b = glb A? Are a, b ∈ A? b. Let B = {x ∈ R | 0 < x^{2} − 4x + 1 < 4}. What is a = lub B? What is b = glb B?

Are a, b ∈ B? c. Let C = {x ∈ Q | x^{2} < 5}. What is a = lub C? What is b = glb C? Are

a, b ∈ C?

‘B‘ Section

1. Let x, y be any real numbers.

a. Show that |x| − |y| ≤ |x − y| and deduce that ||x| − |y|| ≤ |x − y|. b. Show that if x, y > 0, then x < y is equivalent to x^{2} < y^{2}. c. Show that if 0 < x < y, then \sqrt{y} − \sqrt{x} < \sqrt{y − x}.

2. Let a, b be any real numbers. Define max(a, b) and min(a, b) to be the larger and smaller of the two numbers, respectively. (That is, max(a, b) = a if a ≥ b and max(a, b) = b if b ≥ a. Similarly for the minimum.) Show that

max(a, b) = \frac{a + b}{2} + \frac{|a − b|}{2}

and

min(a, b) = \frac{a + b}{2} − \frac{|a − b|}{2}.

3. Show by mathematical induction that

1^{3} + 2^{3} + ⋯ + n^{3} = (1 + 2 + ⋯ + n)^{2}

for all n ≥ 1.

4. Show by mathematical induction that n! ≥ 2^{n} for all n ≥ 4.