Name: 23/; ; aw M MEWS
Math 401
Exam # 1 Friday, September 27, 2013
1. (15 points) Let {an} be a sequence of real numbers, and let A be a real
number. State the deﬁnition of the limit:
n~+oo
lim an = A if and only if.
MR \ 4
I ' \ ‘ if I w .9. 1‘ K a
a
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Name: lemtﬂfd JAM‘WS/tﬁ
Math 401
Exam # 2 Friday, November 8, 2013
1. (15 points) Let f : (1,1) —> R be a function, and let 0 be a point in
(1,1). Deﬁne continuity of f at c:
f is continuous at c if and only if . . .
)wéﬁ < HM”
+w
(Vino (3790
Math 401-1
PS5 assigned Friday Sep. 25th due Friday Oct. 2nd.
1. 2.1: Convergence
1.1. Exercise. Let
n
brk , |r| < 1, b R.
an =
k=0
Prove rigorously that limn an =
b
1r .
2. 2.2: Limit Theorems
1
1
2.1. Exercise. Determine limn ( n ) n if it exists.
So th
Math 401-01
Name:
Quiz 2
2015 Fall
(1) (4 pts)
Let A, B R. Write the negation of each of the following statements:
(a) a A, it is true that a2 B some a0 A s.t. a2 B,
/
(b) For at least one a A, it is true that a2 B a A, a2
/
B,
(c) a A, it is true that a
Final Test: Due December 12, 2014 in class
1. Show that if (1) F1 and F2 are connected sets and (2) F1 F2 is not empty, then
F1 F2 is connected.
2. Suppose that F is connected. Show that F (the closure of F) is also connected.
3. Suppose you are given som
Math 401-1
PS2 assigned Friday Sep. 4th due Friday Sep. 11th.
1. 1.3: Mathematical Induction
1.1. Exercise. By using mathematical induction in detail, prove that the
following holds n N:
n
k2 =
k=1
n(n + 1)(2n + 1)
.
6
1.2. Exercise. Let cfw_sk be a stri
Math 401-1
PS3 assigned Friday Sep. 11th due Friday Sep. 18th.
1. 1.6: Finite and Infinite Sets
1.1. Exercise. Prove that if A and B are countable sets, then A B is
countable.
1.2. Exercise.
(1) Show that R is uncountable.
(2) Show that R \ Q is uncountab
Math 401-01
Name:
Solution to Quiz 1
(1) (3 pts) Write the denition of a prime number. Is 1 a prime number?
Explain why or why not.
The denition is given in notes, or in the textbook. 1 is NOT a
prime because it is not a product of two DISTINCT natural nu
Math 401-1
PS4 assigned Friday Sep. 18th due Friday Sep. 25th.
1. 1.8: Some properties of real numbers
1.1. Exercise. Prove that if a, b 0, then
b2
ab a2 + .
2
2
Moreover, prove that a, b R+ cfw_0,
1
(a2 + b2 ) 2 a + b.
2. 2.1: Convergence
2.1. Exercise.
Math 401-1
1. Running HW due the day of Exam 1
1.1. Exercise. Prove Theorem 1.1.8 from the book, the Pythagorean Theorem.
1.2. Exercise. If A is a set with m elements and B is a set with n elements,
how many elements does A B have? Explain.
1.3. Exercise.
Math 401-01
Name:
Quiz 3
(1) (3 pts)
(a) Circle one:
(i) Course is easier than expected.
(ii) Course is more or less what we expected.
(iii) Course is harder than expected.
(b) Circle one:
(i) HW load and diculty is too light.
(ii) HW load and diculty is
Math 401-1
PS1 assigned Friday Aug. 28th due Friday Sep. 4th.
1. 1.1: Algebra of Sets
1.1. Exercise. Let A, B, C be sets. Prove rigorously showing one subset
direction and the other that
(A B) C = (A C) (B C).
1.2. Exercise. Deduce the second equality of