MATH 242 – Principles of Analysis Problem Set 2 – due: Feb. 7

‘A‘ Section

1. Let x ∈ [−1, 2]. Determine the largest and smallest values of |x − 5|, |x + 5|.

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} + 3b)^{5}.

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

(x^{5} + \frac{3}{x^{2}} )^{3} .

c. Express with no summation: ∑_{k=0}^{n} \binom{n}{k}2^{k}? Explain. d. Express with no summation: ∑_{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 > 0 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 = {y/x | x ∈ A, y ∈ B} is also bounded.

4.

a. Let A = [0, 3) ∩ (2, 5]. What is a = lub A? What is b = glb A? Are a, b ∈ A? b. Let B = {x ∈ R | 0 < x^{2} − 2x + 1 < 1}. What is a = lub B? What is b = glb B?

Are a, b ∈ B?

‘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^{n} < y^{n} for all n ≥ 1.

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}.

(Hint: Consider the positions of a, b on the number line and then “think geometrically.”)

3. Show by mathematical induction that

(1 + x)^{n} ≥ 1 + nx

for all x > −1 and all n ≥ 0.

4. Find a suitable n_{0} and then show by mathematical induction that n! ≥ 5^{n} for all n ≥ n_{0}.