Math 301/ENAS 513, Fall 2007 Midterm Solution
Triet Le October 29, 2007
1. (10 points). Prove that each bounded real sequence has a convergent subsequence. Discuss how this result is also true for bounded complex sequences. You can assume the followi
Math 301 Second Exam Monday, Nov. 9, 2005 Try to give as much detail as possible in your answers. If you use a theorem, state clearly which theorem you are using and show that the hypotheses of that theorem apply. 1. Let X be a metric space with metric d(
Math 301 First Exam Monday, Oct. 10, 2005 Try to give as much detail as possible in your answers. If you use a theorem, state clearly which theorem you are using and show that the hypotheses of that theorem apply. 1. a) State the definition of a Cauchy se
HW1 and HW2 Solution
1
1.1
HW1
Problem 16
Assume Aj , j = 1, 2, 3, 4 , stands for the event that a soldier lost one eye, lost one ear, lost one hand and lost one leg respectively. Then,
P (A1 ) .70 P (A2 ) .75 P (A3 ) .80 P (A4 ) .85. Goal: Minimize P (A1
HW3 Solution
1
1.1
Section 3.1
Problem 3
For each bit, two choices (either 0 or 1) are available. Therefore for a word consisting a string of 32 bits, 232 different possibilities are available.
1.2
Problem 6
This problem is the same as arranging n people
Math 301 First Exam Thursday, Oct. 23, 2008 Try to give as much detail as possible in your answers. If you use a theorem, state clearly which theorem you are using and show that the hypotheses of that theorem apply. 1. a) State the definition of a Cauchy
Math 301 First Exam Thursday, Oct. 23, 2008 Try to give as much detail as possible in your answers. If you use a theorem, state clearly which theorem you are using and show that the hypotheses of that theorem apply. 1. a) State the definition of a Cauchy
Math 301 Final Exam, 2005
Prof. Peter Jones
December 13, 2005
Try to give as much detail as possible in your answers. If you use a theorem,
state clearly which theorem you are using and show that the hypotheses of
that theorem apply.
1. a) State the Heine
Math 301 Final Exam, 2005
Prof. Peter Jones
December 13, 2005
Try to give as much detail as possible in your answers. If you use a theorem,
state clearly which theorem you are using and show that the hypotheses of
that theorem apply.
1. a) State the Heine
Math 301 Second Exam Monday, Nov. 9, 2005 Try to give as much detail as possible in your answers. If you use a theorem, state clearly which theorem you are using and show that the hypotheses of that theorem apply. 1. Let X be a metric space with metric d(
Math 301 First Exam Monday, Oct. 10, 2005 Try to give as much detail as possible in your answers. If you use a theorem, state clearly which theorem you are using and show that the hypotheses of that theorem apply. 1. a) State the definition of a Cauchy se
Math 301/ENAS 513, Homework 1 Solution
Triet Le September 17, 2007
1. Recall: Z = {m - n : m, n N}. Each n N is identified with the expression (n+m)- m for any m N. Addition on Z is defined as: (m-n)+(m -n ) = (m+m )-(n+n ). Let k, m, n N such th
Math 301/ENAS 513: Homework 2 Solution
Triet Le September 21, 2007
1. Show that for any reals x and y with x < y, there are a rational r and an irrational t such that x < r < y and x < t < y. Solution 1. Suppose that x > 0. Let z = y - x > 0 and let
Math 301/ENAS 513: Homework 3 Solution
Triet Le October 3, 2007
1. Let {zn } be a complex sequence. Show that
n
lim zn = z lim Re(zn ) = Re(z) and lim Im(zn ) = Im(z).
n n
Solution 1. Let {zn } be a complex sequence. () : Suppose limn zn = z. Then
Math 301/ENAS 513: Homework 5 Solution
Triet Le October 14, 2007
1. Let K be a fixed natural number and let be a map from N to N such that is 1-to-1 from {(n - 1)K, (n - 1)K + 1, ., nK} onto itself, for all n = 1, 2, 3. Suppose n=0 an converges.
Math 301/ENAS 513: Homework 4 Solutions to Selected Problems
Triet Le October 11, 2007
1. Page 44: 10. Prove that M [0, ) = 0, 1 , where M is the Mandelbrot set. 4 Solution 1. To show [0, 1/4] M [0, ): Suppose c [0, 1/4]. Claim the sequence {zn
Math 301/ENAS 513: Homework 6 Solution
Triet Le
1. Page 66: 2. Determine the coefficients {an } of the power series whose sum is (1 - n=0 z)-2 . Solution 1. Let f (z) = (1 - z)-2 =
n=0
an z n . Then an =
f (n) (0) , n!
where
f (n) (0) = (n + 1)!
Math 301/ENAS 513: HW 7 Soln of Selected Problems
Triet Le
1) Page 78: 5: Solution 1. Let A R C with the usual metric d(x, y) = |x - y|. Pick any x A, then for any > 0, x + i /2 N (x). Hence, N (x) is not a subset of A, for all > 0. And so x is n
Math 301/ENAS 513: HW 8 Solution
Triet Le
Remark 1. Let U be any collection of subsets of a generic set S. Then
c
U
U U
=
U U
U c.
1: If K = {K } is a collection of compact subsets of a metric space (S, d) such that the intersection of every fini
Math 301/ENAS 513: Homework 9 Solution
Triet M. Le
1. Page 89: 8, 10, 11, 13. 15. 2. Page 91: 3,6. 3. Page 94: 2, 3, 6. Page 89: 8: B S being not closed implies B does not contain all its limit points. Let 1 a S be a limit point of B such that a B
Math 301/ENAS 513: Homework 10 Solution
Triet M. Le
Page 104: 2: f is real on R and |f (x)| < M for all x R. Then for any x = y, the mean value theorem implies f (x) - f (y) = f (c), for some c between x and y. x-y Hence, for any > 0, |f (x) - f (y)
Math 301/ENAS 513: Homework 11 Solution
Triet M. Le
Page 107: 3(a): Let P be a polynomial of odd degree. One sees that limx- P (x) = - and limx P (x) = . Let a, b R such that P (a) < 0 and P (b) > 0. Clearly, P is continuous on [a, b]. By the Interm
Math 301/ENAS 513: Homework 12 Solution
Triet M. Le
Page 116: 2: Let f (x) = 0, x0 e-1/x , x > 0.
To show f is infinitely differentiable at x = 0. Clearly, f is infinitely differentiable on (-, 0) (0, ), with 0, x<0 f (n) (x) = -1/x P2n (1/x)e , x