Mathematics 242 – Principles of Analysis Exam 1 – October 5, 2005

Directions: Do all work in the blue exam book. There are 100 possible regular points and 10 possible Extra Credit points.

I. A) (10) State the Axiom of Completeness for the real number system.

B) (10) Show that if s = sup(A) for some A ⊂ R and ε > 0, then there exists some a ∈ A satisfying s − ε < a ≤ s.

C) (10) Let A = ∪_{n=1}^{∞} (0, 1 − \frac{1}{2n}). Explain why A is bounded above and determine sup(A).

II. A) (5) Find

lim_{n→∞} \frac{2n^{2} − 1}{5n^{2} + 2n}

using the Algebraic Limit Theorem.

B) (15) Prove that x_{n} = \frac{2n^{2}−1}{5n^{2}+2n} converges to your limit from part A using the definition.

III. Let (x_{n}) be the sequence defined by x_{1} = 2 and x_{n+1} = \frac{2x_{n}−1}{3} for all n ≥ 1.

A) (15) Using mathematical induction, show that x_{n} ≥ −1 for all n ≥ 1.

B) (15) Using mathematical induction, show that (x_{n}) is strictly decreasing: x_{n+1} < x_{n} for all n ≥ 1.

IV. True-False. For each true statement give a short proof or reason; for each false statement give a counterexample.

A) (10) If A and B are sets of real numbers with the property that a > b for all a ∈ A and b ∈ B, then inf(A) > sup(B).

B) (10) If r is a nonzero rational number and t is an irrational number, then rt is irrational.

Extra Credit (10) Caution: this problem may be “habit forming.” Only attempt after finishing the rest of the exam! Can you find a 1-1, onto function f : R → [0, 1]? If so, describe one by giving a formula or a graph. If not, say why not.