MATH 242 – Principles of Analysis Exam 2 – April 1, 2011

Directions

Do all work in the blue exam booklet. There are 100 possible regular points and 10 Extra Credit Points.

I. A) (20) State and prove the Monotone Convergence Theorem for sequences. (You may

give the proof in the case that the sequence is monotone increasing.)

B) (10) Suppose that x_{n} is a sequence of strictly negative numbers and x_{n+1}/x_{n} ≤ 1 for all n ≥ 1. Show that lim_{n→∞} x_{n} = a exists in R and satisfies a ≤ 0.

II. (20) Suppose {x_{n}} is a sequence such that |x_{n} − 10| < 100 for all n ≥ 1. Show that there exists some number a ∈ [−90, 110] and a subsequence {x_{n_{k}}} such that x_{n_{k}} → a. State any “big theorems” you are using.

III. A) (10) Show using the ε, δ definition that lim_{x→2} x^{2} − x = 2. B) (10) Show using the ε, n_{0} definition that lim_{n→∞} \frac{(−1)^{n}}{n^{2}+1} = 0.

IV. True/False. For each true statement, give a short proof. For each false statement, give a counterexample or reason.

A) (10) If {x_{n}} diverges, then for all a ∈ R, there exist ε_{0} > 0 and n_{0} ∈ N such that |x_{n} − a| ≥ ε_{0} for all n ≥ n_{0}.

B) (10) It is possible to find a function f : R → R that is continuous at c = e, with f(e) = 0, but satisfying f(x) = 1 for all rational x.

C) (10) The sequence defined by x_{1} = 1 and x_{n} = \sqrt{3x_{n−1} + 1} for n ≥ 2 is monotone increasing.

Extra Credit. (10)

Assume that {x_{n}} is a sequence that converges to a. Construct a new sequence {y_{n}} by making y_{n} the average of the first n terms in the original sequence: y_{n} = \frac{x_{1}+x_{2}+···+x_{n}}{n} . Show that {y_{n}} also converges to a.