Mathematics 242 – Principles of Analysis Information on Exam 1 February 15, 2013

General Information

The first exam for the course will be given in class on Friday, February 22. This will be an individual, in-class, closed book exam. I will be happy to hold a late afternoon or evening review session to help you prepare.

What to Expect

The exam will have four or five questions, each possibly with several parts. Some questions will ask for a precise statement of a definition or a theorem we have discussed. I might ask you to give one of the proofs of key results we have covered. The Sample Questions below should give you some idea what they might look like. (The actual exam questions will be different of course!)

Topics to be Covered

The exam will cover the material we have covered since the start of the semester up to and including class on Friday, February 15. This includes all of the material from Chapter 1 and section 2.1 in the text:

1) Sets, set operations (unions, intersections, complements)

2) Functions, the one-to-one and onto properties

3) The Well Ordering Property of N and the technique of proof by mathematical induction. Application to the binomial theorem.

4) Algebraic and order properties of R. Know the proof that \sqrt{2} ∉ Q.

5) Upper and lower bounds, least upper bound and greatest lower bound. The Least Upper Bound Axiom for the real number system: Know the statement and what it means.

6) Consequences of LUB Axiom (know these proofs):

(a) N is not bounded in R.

(b) If ε > 0, then there exists n ∈ N such that 0 < \frac{1}{n} < ε.

(c) If a < b in R, then there is a rational number r ∈ Q such that a < r < b (and hence infinitely many) and also an irrational s ∈ Q^{c} such that a < s < b (and hence infinitely many).

5) Sequences and the definition of convergence.

Practice Questions

Don’t be too concerned about the length of this list. The actual exam will be somewhat less than half the total length of these questions. The idea is to show the range of different types of questions and topics that might be covered.

I. Let A = [1, 5), B = (−1, 2], and C = (−1, 1).

A) What are A ∪ B, A ∩ B, A − B = A ∩ B^{c}? B) What are lub(A), lub(B)? C) Let C = {x · y|x ∈ A, y ∈ B}. Identify lub(C) and prove your claim. D) Let D = {1/x|x ∈ A}. What are lub(D) and glb(D)? Give an explanation how you

know your claims are true. E) Prove that if E is any bounded set of strictly positive real numbers, and F = {1/x|x ∈

E}, then lub(F) = 1/glb(E). F) Is there a one-to-one and onto function f : R → C? If so find one; if not say why not.

II. A) Prove by induction: 2n^{3} + 3n^{2} + n is evenly divisible by 6 for all n ≥ 1. B) Prove by induction: \frac{3^{n}}{n!} > 1 for all natural numbers n ≥ 7.

III. Saying a function f : R → R is strictly increasing means: for all real numbers x and y satisfying x < y, the inequality f(x) < f(y) holds.

A) Give the negation of the statement f is strictly increasing in the same sort of form as the definition above. (What does it mean for that statement to be false? Just saying “f is not strictly increasing” is not an acceptable answer!)

B) Give the contrapositive of the statement: “If f is one-to-one, then f is strictly in-

creasing.” Is that a true statement? C) Consider f : R → R defined by f(x) = x^{2}. What is wrong with the following

reasoning? If x < y, then x^{2} < y^{2}; therefore f(x) = x^{2} is a strictly increasing function.

IV. A) State the Least Upper Bound Axiom for the real number system. B) Show that for any real number number y > 1, there exists a natural number n such

that 1 + \frac{1}{n} < y. C) Let n represent a natural number and let A = ∩_{n=1}^{∞} [1, 1 + \frac{1}{n}]. Is A the empty set?

Why or why not? D) Let B = ∪_{n=1}^{∞} (0, 1 + \frac{1}{n}). What is lub(B)? Explain.

V. A) Prove that if a < b are any real numbers, then there is some r ∈ Q such that a < r < b. B) Show that if r is a rational number, then \sqrt{2} + r is an irrational number. (Argue by

contradiction.) C) Show that if a < b are any real numbers, then there is an irrational number s with

a < s < b.

VI. A) Show using the definition that the sequence x_{n} = \frac{n+1}{3n+2} converges to \frac{1}{3}. B) Show using the definition that the sequence x_{n} = 1 + (−1)^{n} does not converge.