Principles of Analysis

Homework Assignment #6

Due Wednesday, March 13, 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 2.5
Problems:   2a, 2b, 4a, 4c, 7, 13

Extra Credit Problem: Given any sequence {x_n}. Let S be the set of all possible subsequences of {x_n}. So each element of S is a particular subsequence {x_n_k}. Is S a countable or uncountable set? Provide a proof to receive credit.

Hints: For problem #2b, [n/5] means the greatest integer less than or equal to n/5. So [2/5] = 0 while [6/5] = 1. In problem #2a and #2b, give the index sequence n_k you use to generate your subsequences. Problems #4 and #13 ask you to come up with sequences with certain properties. You do not have to come up with particular formulas to your sequences (like x_n = 1/n) but the sequences you do come up with should have a clear pattern to them. (In other words, it should be clear to the reader how to continue your sequence beyond the terms listed.) These problems are designed to get you to think clearly about the definition of convergence and the definition of a subsequence.

Section 3.1
Problems:   2a, 2b, 2h, 2m, 3, 6

Hint: For problem #2, you find the limit and then prove it exists using the epsilon-delta definition. Good examples to refer to are those from your class notes and those in the textbook. For problem #3, try drawing a graph.