Mathematics 242 – Principles of Analysis Information on Exam 2 March 20, 2013

General Information

The second hour exam for the course will be given in class on Friday, April 5. This will be an in-class, closed book exam. You may use a calculator, but no other electronic devices are allowed and no stored text or other saved or online information can be consulted during the exam. I will be happy to hold a late afternoon or evening review session to help you prepare. Late afternoon or evening times are possible on Wednesday, April 3 or Thursday, April 4.

Topics to be Covered

The exam will cover the material we have covered since the last exam, up through and including class on Friday, March 22. (This is the content of Problem Sets 4,5,6, plus the material on properties of continuous functions on closed intervals.) (Of course, all the material from the first exam about the real number system, least upper bounds, completeness, and the technique of proof by mathematical induction is relevant here too!)

1) Sequences and convergence.

2) The Limit Theorems for sequences.

3) The Monotone Convergence Theorem for sequences and its consequences (including using it to analyze sequences defined inductively).

4) Subsequences and the Bolzano-Weierstrass Theorem.

5) The definition of the statement lim_{x→c} f(x) = L, variants such as one-sided limits, consequences, limit theorems, techniques for computing limits, including use of sequences to detect when limits do not exist, or to prove that they do.

6) The definition of continuity and its consequences (including properties of removable, jump and infinite discontinuities), key properties of continuous functions on a closed interval: The Intermediate Value and Extreme Value Theorems.

What to Expect

The format will be similar to that of the first exam. The exam will have four or five questions, each possibly with several parts. Some questions will ask for a precise statement of a definition or a theorem we have discussed. Be prepared to give careful statements of

1) The definition of convergence for a sequence, and the statement lim_{x→c} f(x) = L.

2) The Bolzano-Weierstrass Theorem.

3) The Extreme Value and Intermediate Value Theorems for continuous functions on closed intervals.

Also know and be able to give these proofs:

1) Part (a) of the Limit Theorem for sequences (Theorem 2.2.5) (limit of a sum is the sum of the limits).

2) The Monotone Convergence Theorem for sequences (Theorem 2.3.3 in our text).

3) The proof that if f is continuous at c and {x_{n}} is a sequence convering to c with x_{n} in the domain of f for all n, then f(x_{n}) converges to f(c) (Theorem 3.4.13 in text).

Practice Questions

Don’t be concerned about the length of this list. The actual exam will be roughly the same length as Exam 1. The idea is to show the range of different types of questions and topics that might be covered.

I. Give an example of each of the following, or give a short proof that there are no such examples: A) A convergent sequence with all strictly negative terms whose limit is 1. B) A sequence {x_{n}} such that {x_{n}^{2}} converges but {x_{n}} does not. C) Sequences {x_{n}} and {y_{n}} such that lim x_{n} = +∞ but {x_{n}y_{n}} converges to a finite

limit. D) A monotone increasing sequence {x_{n}} that has no convergent subsequence. E) A continuous function f : R → R such that f(x) = 0 if x ∈ Q, but f(π) = 3. F) A function f : [a, b] → R with f(a) = 3, f(b) = −1, but such that f(x) = 0 has no

solution in [a, b]. G) A function f and a c in the domain of f such that lim_{x→c^{+}} f(x) = lim_{x→c^{−}} f(x), but

f is not continuous at c. H) A function that is continuous at x = 0 such that f(x) ≥ 0 for all x ≠ 0 but f(0) =

−.00000001 I) A sequence with subsequences converging to 7 different limits (and no others). J) A function that is continuous at c for which there is a sequence x_{n} → c in the domain

of f with {f(x_{n})} unbounded. K) A continuous function f on a closed interval [a, b] such that M = lub{f(x) | x ∈ [a, b]}

but M ≠ f(x) for any x ∈ [a, b].

II. Show using the definition that the sequence x_{n} = \frac{n+1}{3n+2} converges to \frac{1}{3}.

III. A) Show that if x_{n} → a and y_{n} → b, then x_{n} + y_{n} → a + b. B) Find lim_{n→∞} \sqrt{n}(\sqrt{9n + 1} − 3\sqrt{n}).

IV. Let \{x_{n}\} be the sequence defined by x_{1} = 1 and x_{n} = \frac{3x_{n−1}+1}{5} for all n ≥ 2. A) Using mathematical induction, show that x_{n} ≥ 1/2 for all n ≥ 1. B) Using mathematical induction, show that {x_{n}} is monotone decreasing. C) Use part B to show that the sequence x_{n} converges to 1/2. D) An alternate proof that this sequence converges: Using mathematical induction, show

that |x_{n} − \frac{1}{2} | = \frac{1}{2} · (\frac{3}{5})^{n−1} for all n ≥ 1 and use that to show that x_{n} → 1/2.

V. Prove the following statements using the ε − δ definition of functional limits: A) lim_{x→3} x^{2} − 2x + 4 = 7. B) lim_{x→e}[x] = 2 ([x] is the greatest integer function) C) lim_{x→2} \frac{1}{x^{2}} = \frac{1}{4}.

VI. Let f(x) = x^{7} − 5x + 1. A) Prove that f(x) is continuous at all c ∈ R. B) State the Intermediate Value Theorem. C) Prove that f(x) = 0 has a solution in the interval (0, 1) and another in the interval

(1, 2).

VII. True/False and prove/give a counterexample. Let f be continuous on [0, 1] and let f(x) ∈ [0, 1] for all x ∈ [0, 1]. Then f(x) = x^{2} for some x ∈ [0, 1]. (Hint: IVT)

More Challenging Problems

You should expect about 10 of the 100 points possible on the exam to be a problem that is more challenging (to separate the A’s from the A-’s). I will also give an Extra Credit problem. This is indicative of the level for those components on this exam.

IX. In all parts of this problem, {x_{n}} is a bounded sequence. Let S be the set of all real numbers s such that there exists a subsequence {x_{n_{k}}} converging to s. (S is the set of cluster points of the sequence, as defined in Problem Set 5.)

A) Show that S is bounded above and below.

The number t = lub(S) is sometimes called the limit superior of the sequence {x_{n}}, written

t = lim sup x_{n}.

B) What is lim sup (−1)^{n} + \frac{1}{n}? C) Show that if ε > 0 and t = lim sup x_{n}, then there are only finitely many terms of the

sequence x_{n} with x_{n} > t + ε.