Mathematics 242 – Principles of Analysis Exam 2 – April 5, 2013

Directions

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

I. A) (15) Show using the ε, n_{0} definition that

lim_{n→∞} \frac{5n + 1}{3n + 3} = \frac{5}{3}.

B) (15) Show using the ε, δ definition that

lim_{x→1} \frac{5x + 1}{3x + 3} = 1.

II. 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) Let {x_{n}} be the sequence defined by x_{1} = 1 and

x_{n+1} = \frac{5x_{n} + 3}{16}

for all n ≥ 1. Does this sequence converge? Why? If it does, what is the limit?

III. (10) Let x_{n} = sin(n) − cos(n) (the sequence of values of sin(x) − cos(x) at the angles x given by a whole number n ≥ 1 in radian measure). Show that there exists a subsequence x_{n_{k}} = sin(n_{k}) − cos(n_{k}) that converges. State any “big theorems” that you are using.

IV. Give an example, or give a reason why there can be no such examples:

A) (10) A function f that is continuous at all real c, with f(x) = 3 for all x ∈ Q, but f(\sqrt{2}) = 4.

B) (10) A function f that is continuous and bounded on an open interval (a, b) with M = lub{f(x) | x ∈ (a, b)}, but such that f(x) ≠ M for all x ∈ (a, b).

C) (10) A continuous function on [0, 1] with f(0) = 4, f(1) = −2, but f(x) ≠ 0 for all x ∈ (0, 1).

Extra Credit. (10) A function is said to be right-continuous at c if the one-sided limit lim_{x→c^{+}} f(x) exists and equals f(c). True or False (and give a proof or a counterexample): If f is right-continuous at all c ∈ [a, b] for some closed interval [a, b], then {f(x) | x ∈ [a, b]} is a bounded subset of R.