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 bou
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 t
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 t
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
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 t
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 t
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
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
Pr
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
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
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
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) (
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.
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-
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
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
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,
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 [
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
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
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 spa
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 point
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 betwe