I will try to update this page every lecture, to announce the due dates for the assignments from each section covered. The problems listed below are tentative (and may change!!) until these due dates have been announced. Always recheck that you have done the problems actually assigned, before submitting them!
Assignments
(Numbers refer to 4th Edition. If you have an earlier edition, please compare with the 4th edition to see what problems you need to do.)
EXTRA CREDIT 1: (Due Mon, Feb 5)
What was the mistake in the inductive “proof” done in class to show any n objects were the same?
HW1 (Due Wed, Feb 7)
Sec 1.1: 10, 16, 22
Problem A0: Let A1={3k: k ∈ N}, A2={3k-1: k ∈ N}, A3={3k-2: k ∈ N}. Also, let E be the set of all even natural numbers.
Give the intersection of E with each of A1, A2, and A3. State each as a set of the form S={formula involving k, k ∈ N}.
What is the union of A1, A2 and A3?
Watch Video: The Big Bang of Numbers
Sec 1.2: #1,5,7,11,14
HW 2 (Due Wed, Feb 14)
Sec 2.1: # 7,8(b),9
Sec 2.2: # 5, 6(a), 16, 17
Problem A1: First show that if x is in V_ε(a) and y is in V_ε(b), then x+y has to be in V_(2ε)(a+b). Next, by counterexample, show that x+y does not have to be in V_ε(a+b). (Give actual values of a, b, x, y and ε for this counterexample)
HW 3 (Due Wed, Feb 21)
Sec 2.3: # 1, 4(Justify!), 8, 9, 11
(NOTE: Use only arguments from this section, not Sec 2.4)
Sec 2.4: # 2, 3, 5, 19
HW 4 (Due Wed, Feb 28)
Sec 1.3: # 4, 12, 13 (hint at back) Also, #1 through 6 from: Cardinality Problems.
True/False self-assessment test on Blackboard (about absolute value and neighborhoods). Access questions here and do test on Blackboard.
EXTRA CREDIT: Due Date Mon, Mar 5
You are given an ubiased coin for which the probability of getting a head (or a tail) in any toss is exactly 1/2. You toss it repeatedly, to generate an infinite sequence (x_n) whose terms are given by
x_n=(number of heads in n tosses)/n
(i.e. x_1=(number of heads in 1st toss)/1, x_2=(number of heads in first 2 tosses)/2, etc).
Say whether or not Lim x_n = 1/2 in the sense of definition 3.1.3, justifying your answer.
HW 5 (Due Wed, Mar 14)
Sec 3.1 #5(b),(d), 7, 18 (NOTE: Only use the definition in problems from this section, not theorems from Sec 3.2)
Sec 3.2 # 4,7,9,15, 22
EXTRA CREDIT (submit separately): Suppose (x_n) is a convergent sequence with limit L. What can you say about the convergence or divergence of the sequence (log(x_n))? Prove your results.
CELEBRATE PI DAY! Read my New York Times article on Pi Day from 2015. Also, watch the related video here.
HW 6 (Due Wed, Mar 28)
Sec 3.2 # 5 (Use only results from Sec 3.1 and 3.2, not 3.3 or 3.4)
Sec 3.3 # 3,5,7,9, 12(c)(d)
Use the break to fill in the gaps: material you may not have mastered yet.
HW 7 (Due Wed, Apr 4)
Sec 3.4: # 3,4,9,12,14
Sec 3.5 # 2(a), 3(b), 4, 5, 9
HW 8 (Due Wed, Apr 11)
Sec 3.5 #12, 13
Sec 3.7 # 3(b), 4,5, 10, 11, 12,13, 14
EXTRA CREDIT (submit separately): Given a series S=Sum(a_n), one can define a subseries S’=Sum(a_n_k). Prove or Disprove: Every positive p-series, p>0, always has a convergent subseries.
EXTRA CREDIT (Due Mon, Apr 23)
King Lear Problem: We know that any angle can be bisected using straight edge and compass. (a) How would you apply this bisection technique repeatedly to approximately trisect the angle? (b) By considering the limit of an infinite series, prove that you converge to the trisected angle as the number of repetitions approaches infinity.
HW 9 (Due Wed, Apr 25)
Sec 4.1 #6,7 (use definition, not sequential criterion),10(a),12(d), 14
Sec 4.2 # 4,5,9, 11(d)
HW 10 (Due Wed, May 2)
Sec 5.1 # 3, 7,11,12,13
Sec 5.2 #3,5
Sec 5.3 # 4,6
EXTRA CREDIT (Due Mon, May 7)
In a previous extra credit problem, nobody was able to prove that if (x_n) is a convergent sequence converging to L (where x_n, L are both positive) then the sequence log(x_n) converges to log(L). Now that we’ve covered Sec 6.2, can you prove it? (To get credit, I want you to use the definition of the limit for sequences.)
HW 11 (Due Wed, May 9)
Sec 6.1 # 2,4,9,10,13
Sec 6.2 #6,8,9,17
EXTRA CREDIT (Due Mon, May 14)
Sec 11.1 # 5,6,8
Problem L1: True or false (justify in each case):
(1) If A is closed, then removing a single point from A will always give a set that is no longer closed.
(2) If A is open, then removing a single point from A will always give a set that is no longer open.