Solucionario rudin

Supplements to the Exercises in Chapters 1-7 of Walter Rudin’s Principles of Mathematical Analysis, Third Edition
by George M. Bergman This packet contains both additional exercises relating to the material in Chapters 1-7 of Rudin, and information on Rudin’s exercises for those chapters. For each exercise of either type, I give a title (an idea borrowed from Kelley’s General Topology), anestimate of its difficulty, notes on its dependence on other exercises if any, and sometimes further comments or hints. Numbering. I have given numbers to the sections in each chapter of Rudin, in general taking each of his capitalized headings to begin a new numbered section, though in a small number of cases I have inserted one or two additional section-divisions between Rudin’s headings. My exercisesare referred to by boldfaced symbols showing the chapter and section, followed by a colon and an exercise-number; e.g., under section 1.4 you will find Exercises 1.4:1, 1.4:2, etc.. Rudin puts his exercises at the ends of the chapters; in these notes I abbreviate ‘‘Chapter M, Rudin’s Exercise N ’’ to M : R N. However, I list both my exercises and his under the relevant section. It could be arguedthat by listing Rudin’s exercises by section I am effectively telling the student where to look for the material to be used in solving the exercise, which the student should really do for his or her self. However, I think that the advantage of this work of classification, in showing student and instructor which exercises are appropriate to attempt or to assign after a given section has beencovered, outweighs that disadvantage. Similarly, I hope that the clarifications and comments I make concerning many of Rudin’s exercises will serve more to prevent wasted time than to lessen the challenge of the exercises. Difficulty-codes. My estimate of the difficulty of each exercise is shown by a code d: 1 to d: 5. Codes d: 1 to d: 3 indicate exercises that it would be appropriate to assign in anon-honors class as ‘‘easier’’, ‘‘typical’’, and ‘‘more difficult’’ problems; d: 2 to d: 4 would have the same roles in an honors course, while d: 5 indicates the sort of exercise that might be used as an extra-credit ‘‘challenge problem’’ in an honors course. If an exercise consists of several parts of notably different difficulties, I may write something like d: 2, 2, 4 to indicate that parts (a) and (b)have difficulty 2, while part (c) has difficulty 4. However, you shouldn’t put too much faith in my estimates – I have only used a small fraction of these exercises in teaching, and in other cases my guesses as to difficulty are very uncertain. (Even my sense of what level of difficulty should get a given code has probably been inconsistent. I am inclined to rate a problem that looks straightforwardto me d: 1; but then I may remember students coming to office hours for hints on a problem that looked similarly straightforward, and change that to d: 2.)
The difficulty of an exercise is not the same as the amount of work it involves – a long series of straightforward manipulations can have a low level of difficulty, but involve a lot of work. I discovered how to quantify the latter some yearsago, in an unfortunate semester when I had to do my own grading for the basic graduate algebra course. Before grading each exercise, I listed the steps I would look for if the student gave the expected proof, and assigned each step one point (with particularly simple or complicated steps given 1⁄2 or 11⁄2 points). Now for years, I had asked students to turn in weekly feedback on the time their studyand homework for the course took them; but my success in giving assignments that kept the average time in the appropriate range (about 13 hours per week on top of the 3 hours in class) had been erratic; the time often ended up far too high. That Semester, I found empirically that a 25-point assignment regular kept the time quite close to the desired value. I would like to similarly assign...