Mathematics 242 – Principles of Analysis Exam 1 February 22, 2013

Directions: Do all work in the blue exam booklet (do not place any work that you want considered for credit on this sheet). There are 100 regular and 10 extra credit points.

I. A) (10) Let

A = ⋃_{n=1}^{∞} (\frac{1}{n}, 2 − \frac{1}{n})

Explain why A is bounded and determine upper and lower bounds for A. B) (10) Define: The real number a is a least upper bound of A ⊂ R, and state the Least

Upper Bound Axiom for R. C) (10) Let A be a bounded subset of R and let B = {4 · x − 3 | x ∈ A}. What can be

said about lub(B)? Prove your assertion.

II. (20) Let x_{n} be the sequence defined by the rules x_{1} = 1 and x_{n+1} = \frac{1}{3}x_{n} + 1 for all n ≥ 1. Show by mathematical induction that

x_{n} = \frac{1 − \frac{1}{3^{n}}}{1 − \frac{1}{3}} for all n ≥ 1.

III. Let x_{n} = \frac{5^{n}}{7^{n}+3} for all natural numbers n ≥ 1. A) (10) Determine lim_{n→∞} x_{n} intuitively. B) (20) Use the ε, n_{0} definition of convergence to prove that {x_{n}} converges to the number

you identified in part A.

IV. True-False. For each true statement, give a short proof or reason. For each false statement give an explicit counterexample. A) (10) Let A be a bounded set of real numbers and let a = lub(A). Then for each ε > 0,

there exists x ∈ A such that a − ε < x < a. B) (10) If r is a nonzero rational number and s is a nonzero irrational number, then r/s

is irrational.

Extra Credit (10) Is it possible to produce a sequence x_{n} whose terms include all the rational numbers p/q with p, q ∈ Z and p, q > 0? If so, give an indication how to construct such a sequence. If not, give a reason why there cannot exist such a sequence.