Principles of Analysis

Homework Assignment #3

Due Wednesday, February 6, START of Class

Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to Bilodeau and Thie's book, the required text for the course. You should write up solutions neatly to all problems, paying particular attention to your arguments and proofs. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be your own work.

Section 1.4
Problems:   7, 10, 18

Hint: For problem #7, use the Binomial Theorem and find the correct "k" index in the sum which yields the given term. You will need a calculator to calculate the coefficients. For problem #10, make a judicious choice for x and y in the Binomial Theorem and you can prove each formula quickly. Note that part a) shows that the sum of the nth row of Pascal's Triangle is 2^n. For problem #18, you should prove the theorem by induction although the result follows from #10a (why?).

Section 1.5
Problems:   2, 5, 9 (parts a, b, c), 12 (part a)

Hint: For problem #5, show that lub B <= lub A + k and then show that lub A + k <= lub B. These can be proven using the definition of the least upper bound for A and B. The proof for the glb statement is similar. For problem #9, try making up examples to start with. If the statement is true, you must prove it. If the statement is false, come up with a counterexample.

Additional Problem:
Read Section 1.5 carefully. Using the proof in the text, but in your own words, prove Corollary 1.5.2, which states that a nonempty set of real numbers bounded below has a greatest lower bound.