Mathematics 242 – Principles of Analysis Makeup 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) Define: The real number a is a least upper bound of A ⊂ R, and state the Least

Upper Bound Axiom for R. B) (10) Let

A = ⋂_{n=1}^{∞} (1 + \frac{1}{n}, 4 − \frac{1}{n})

Explain why A is bounded and determine the least upper and greatest lower bounds for A.

C) (10) Let A be a bounded subset of R and let B = {−2x + 1 | x ∈ A}. What can be said about glb(B)? Prove your assertion.

II. (20) Let x_{n} be the sequence defined by the rules x_{1} = 1 and x_{n+1} = −\frac{2}{5}x_{n} + 1 for all n ≥ 1. Show by mathematical induction that

0 ≤ x_{n} ≤ 1 for all n ≥ 1.

III. Let x_{n} = \frac{5n^{4}}{n^{4}+3n} 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) If A and B are two nonempty bounded sets of real numbers and lub(B) > lub(A),

then there exists a y ∈ B such that y > x for all x ∈ A. B) (10) If r is a nonzero rational number and s is a nonzero irrational number, then rs

is irrational.

Extra Credit (10) Is it possible to produce a sequence x_{n} whose terms include all the positive and negative integers? If so, give an indication how to construct such a sequence. If not, give a reason why there cannot exist such a sequence.