MATH 323: Real Analysis I

TEXT: Stephen Abbott, Understanding Analysis, Springer-Verlag, 2001.

INSTRUCTOR: David Lantz

WEB ADDRESS: This page has the web address:

http://math.colgate.edu/math323/dlantz/realsfdi.html
I encourage you to bookmark it on your computer for later use.

HOMEWORK: Due dates will be specified.

  1. Page 17 (Ch 1, sec 3): 1.3.3(a), 1.3.4, 1.3.5, 1.3.6, 1.3.7 (easy), 1.3.8 (very easy), 1.3.9; and page 27 (Ch 1, sec 4): 1.4.2(c), 1.4.4. (Solutions)
  2. Page 43 (Ch 2, sec 2): 2.2.1, 2.2.2, 2.2.6, 2.2.7, 2.2.8. (Solutions)
  3. Page 49 (Ch 2, sec 3): 2.3.1, 2.3.2 (For (b), assume x is not 0.), 2.3.3 (If a<b<c, then |b|<max{|a|,|c|}.), 2.3.4 (Assume not, and take as epsilon half the distance between the "limits". The Triangle Inequality is helpful.), 2.3.6, 2.3.8, 2.3.9, 2.3.10, 2.3.11 (After choosing N large enough to make the xn's close to their limit x, choose N1 still larger so that
    (|x1-x| + ... + |xN-x|) / N1
    is small.). (Solutions)
  4. Page 54 (Ch 2, sec 4): 2.4.2 (For (a), note x2-4x+1<0 is equivalent to (x-2)2<3.), 2.4.4, 2.4.5(a) (Use (xn-21/2)2>0 to show xn+1>21/2.); and page 58 (Ch 2, sec 5): 2.5.3. (Solutions)
  5. Page 61 (Ch 2, sec 6): 2.6.1, 2.6.3, 2.6.4; and page 68 (Ch 2, sec 7): 2.7.4 (easy), 2.7.5, 2.7.9, 2.7.10. You may use among your examples the fact that the sum of 1/n2 converges. (Solutions)
  6. Page 82 (Ch 3, sec 2): 3.2.1, 3.2.2, 3.2.3, 3.2.7, 3.2.9, 3.2.12 (and read 3.2.8); and page 87 (Ch 3, sec 3): 3.3.1, 3.3.3, 3.3.4, 3.3.5, 3.3.7, 3.3.9. (Exercise 3.3.6 is an interesting fact, but we probably won't use it.) (Solutions)
  7. Page 108 (Ch 4, sec 2): 4.2.1(a), 4.2.6, 4.2.9; and page 113 (Ch 4, sec 3): 4.3.2, 4.3.3, 4.3.5, 4.3.9 (For (b), verify that, if m> n, then |ym-yn| is at most |y2-y1|(cn-1/(1-c)). For the first part of (c), assume BWOC f(y) is different from y, and use the Triangle Inequality on |f(y) - f(yN) + f(yN) - y|.) (Solutions)
  8. Page 119 (Ch 4, sec 4): 4.4.2, 4.4.4, 4.4.8 (Note f is uniformly continuous on [0,b+1].), 4.4.9; and page 124 (Ch 4, sec 5): 4.5.3, 4.5.7 (Set g(x)=f(x)-x.). (Solutions)
  9. Page 136 (Ch 5, sec 2): 5.2.1, 5.2.2, 5.2.4; and page 143 (Ch 5, sec 3): 5.3.1, 5.3.2, 5.3.5, 5.3.7 (For the last part of (b), evaluate g' at 1/(k pi) for each natural number k.), 5.3.11. (Solutions)
  10. Page 190 (Ch 7, sec 2): 7.2.2, 7.2.3, 7.2.4, 7.2.6 (Hint: Use the partition of [a,b] into n subintervals of equal length and apply Exercise 7.2.4(a).); and page 193 (Ch 7, sec 3): 7.3.1 (Typo: The function should be f, not h.), 7.3.4(c), 7.3.5. (Solutions)
  11. Page 198 (Ch 7, sec 4): 7.4.1, 7.4.2, 7.4.4 (For (d) in particular, go to the definition of integral.), 7.4.5 (In the proof of (b), show L(f,P) + L(g,P) is at most U(f+g,P).) (Solutions)
  12. Page 201 (Ch 7, sec 5): 7.5.1, 7.5.2, 7.5.4 (In (c), integrate 1/t from c to cx by the substitution u=t/c.), 7.5.7, 7.5.9. (Solutions)

TESTS: There will be three mid-term exams, on Tuesday evenings, 7:30-9:30 p.m., on the dates listed below, in our usual classroom, McGregory 201F. A cumulative final exam will be given during the officially scheduled final exam period for this course. If it is impossible for you to take one of the mid-terms during the scheduled period, please see me to set a time to take it early, ahead of the scheduled period. The date and time for the final cannot be altered; please consult the final exam schedule before making travel plans.

GRADING: Homework will be worth one-fifth of the course grade, and each exam (including the final) will be worth one-fifth of the grade.

SIGNIFICANT DATES:

SUGGESTIONS:

  1. If you see an error at the board, or if you do not understand something, stop me and ask (even if everyone else seems to understand).
  2. Doing the homework, regularly and thoughtfully, is absolutely essential to your success in the course.

STUDY AIDS AND GRAPHICS:

Revised: December 10, 2003. Questions to: dlantz@mail.colgate.edu
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