Math 242 Homework and Test Information
Late penalty : 15% per day.
Homework #1. Due: Monday, Jan 29 in class.
A: Write up a detailed proof of Fermat's Theorem.
B: Fill in the details of the proof of Proposition 1.2 (page 30); show that the function (f x g)(s,t)=(f(s),g(t))
is indeed a bijection of S x T with N x N.
C: Prove that the set of all matrices with rational entries is countable, and the set of all matrices with real
entries is uncountable.
From the text, Chapter 1: 11b, 14, 19, 20, 22, 23
From the text, Chapter 2: 23
End of homework #1.
Homework #2. Due: Feb 11
(1) Prove that the limit of the sequence a_n = (2n^2 +1) / (sqrt(n^4 +2)) is 2 using the epsilon-delta definition.
(2) Recall the uniform sequence on [0,1]; a_1 = 1/2, a_2 = 1/4, a_3 = 3/4, a_4=1/8, a_5 = 3/8, ....
(i) Show that this sequence is divergent.
(ii) Calculate the lim inf and lim sup.
(iii) Pick any number a between the lim inf and lim sup you found in part (ii). Then find a subsequence that converges to a.
(3) Show that if lim inf = lim sup = a for some sequence a_n, then a_n has limit a.
(4) Consider the following recursively defined sequence, a_(n+1) = (a_n)^3 + (1/2)a_n, n=1,2,3,... with |a_0|< 1/sqrt(2) but otherwise arbitrary
Show that this is a convergent sequence and find the limit. (Hint: Use induction! First show that 0 < |a_n| < 1/sqrt(2) for all n and use
the "important Theorems" from the text about sequences).
(5) Prove part (4) of Proposition 3.2 (page 77).
(6) Problem 7, Chapter 3
(7) Chapter 4; 1c,d,e
(8) Chapter 4; 4, 5
(9) Chapter 5; 1, 2
(10) Let E be the set of end points of the Cantor set construction; (0, 1, 1/3, 2/3, 1/9, 2/9, ...). Show that the
closure of E is the Cantor set.
End of homework #2.
Notes on the Cantor set.
Homework #3. Due: Wed March 11
Problem A: Prove that limit x --> 1 of f(x) = x^3 + x^2 + x + 1 is 4 using the epsilon-delta definition.
Chapter 6: 10, 12
Chapter 7: 11, 17, 20
Problem B: Provide a complete proof:
f: R^n --> R^n is continuous iff f^{-1}(B(x;r)) is open for every ball
B(x;r) contained in its domain.
(cf. Remark 6.3. You will use Theorem 6.3.) This is an 'easy' proof; concentrate in writting the proper structure of the proof.
Problem C: Find the limits; (i) lim x--> 0 (1-cos(x^2))/(x^3 sin(x)), (ii) lim x--> infinity (1-(1/x))^x.
Problem D: (a) Show that sqrt(|x|) is uniformly continuous.
(b) Show that x^3 is not uniformly continuous.
(c) Show that x^3 is uniformly continuous on [a,b], any a < b finite.
End of Homework #3
Homework #4. Due: Wednesday April 1 (no joke)
Chapter 8 : 2, 7, 13, 23
Chapter 9 : 5 (Hint: Let e>0 and consider the sets Un={x : |f(x)-fn(x)| < e }, 18
Chapter 10 : 7
A : Prove that int_0^x 1/(1+t^2) dt = sum_0^infty [(-1)^nx^{2n+1}/(2n+1)] for |x|<1.
(Here, int_a^b means integral from a to b, and sum_a^b means the sum from n=a to n=b, and infty means infinity.)
B : Suppose that the series sum_0^infty a_n converges. Prove that the function f(x) = sum_0^infty (a_n x^n) defines a continuous function on (-1,1). What more must you assume in order that f(x) be differentiable?
End of homework #4.
Last updated: March 26