Math 3610.001 Homework 3
Due September 27th, 2013
Mance
1. Find an example of a sequence of real numbers satisfying each set of properties.
(a) Cauchy, but not monotone
(b) Monotone, but not Cauchy
(c) Bounded, but not Cauchy
2. A sequence (sn ) is said t
Math 3610.001 Homework 4
Due October 5th, 2013
Mance
1. Problem 14.11
2. Consider the formula
xn
f (x) = lim
.
n 1 + xn
Let D = cfw_x : f (x) R. Calculate f (x) for all x D and determine where
f : D R is continuous.
3. Dene f : R R by f (x) = 5x if x is r
MATH 3610.001 HOMEWORK 4
Problem 1. 14.11
Proof. (a) Each Ak is the nine union of closed sets, hence closed. Thus C is an
intersection of closed sets, hence closed. Certainly C is bounded. Hence C is
compact
(b) in base 3,
A1 = cfw_0.a1 a2 a3 . : a1 cfw_0
MATH 3610.001 HOMEWORK 4
Problem 1. .
1
(a) | sinn 0| n , so given > 0, letN = 1/
n
n2
nn
n
n
(b) n! = n(n1)(n1)(2)(1) n2 3n+2 n2 3n =
1
n3
2
1
n
2
=
2
n
when n 6 So
by excercise 16.2 (c) lim n = 0.
n!
1
(c) Suppose that x = 0. Since |x| < 1, |x| = 1+y fo
MATH 3610.005 HOMEWORK 7
(1) If P (x) is divisible by (xa)2 , then p(x) = (xa)2 q(x) for some polynomial
q(x). Then by the product rule,
p (x) = (x 2)2 q (x) + 2(x a)q(x) = (x a)[(x a)q (x) = 2q(x)].
Thus x a divides p (x)
(2) Suppose f is a dierentiable
MATH 3610.005 HOMEWORK 8
(1) 26.6
f (x) = x on [0, 1] satisess (i) and (ii). f (x) = |x| on [1, 1] satises (i)
and (iii). f (x) = x for 0 < x 1 and f (0) = 1 satises (ii) and (iii). In
each case, there is no point c in the relevant interal for which f (c)
MATH 3610.005 HOMEWORK 9
k
(1) Give > 0, M such that x > M f (x) > |k| . Then for x > M, | f (x) | < ,
k
so limx f (x) = 0.
(2) Suppose limx f (x) = L and let > 0. Then M > 0 such that
1
1
|f (x) L| < whenever x > M . But then 0 < y < M , where x = y . Th
MATH 3610.005 HOMEWORK 10
if x Q
if n R \ Q
5x
x2 + 6
(2) Let P = a, b. Then
(1) Dene f (x) =
b
f = U (f ) U (f, p) M (b a)
m(b a) L(f, P ) L(f ) =
a
(3) Suppose f (c) > 0 for some c [a, b]. Then by Excercise 21.13 > 0 and
a neighborhood U of c such that
Math 3610.001 Homework 2
Due September 20th, 2013
Mance
a
1. Let S = 10n : a Z, n N cfw_0 . Prove that S is dense in R. Note: A
consequence of this fact is that every real number has at least one decimal
expansion.
2. Let y be a positive real number. Prov
Math 3610.001 Homework 1
Due September 13th, 2013
Mance
1. Determine if the following sequences converge or diverge. If they converge, nd
their value. Prove that your answer is correct.
(a) lim sin n
n
2
(b) lim n
n!
(c) For |x| < 1, limn xn
(d) lim cos n
Math 3610.001 Homework 10
Due 11-15-13
Mance
For this homework you may use any dierentiation formulas you know
from calculus.
1. Give an example of a function f : [0, 1] R that is not integrable on [0, 1],
but f 2 is integrable on [0, 1].
2. Let f be inte
Math 3610.001 Homework 12
Optional
Mance
For this homework you may use any dierentiation formulas you know
from calculus.
1. 35.3 through 35.8
2. 35.10
3. 35.11 and 35.12
4. 35.17
5. 35.19
6. 36.3
7. 36.9
8. 36.10
9. 36.13
10. 36.15 and 36.16
11. 36.17
1
Math 3610.001 Homework 6
Optional homework
Mance
1. 23.4
2. 23.5
3. 23.11
4. 23.15
5. 25.5
6. 25.11
7. Determine where the modied Dirichlet function described in example 21.9 is
dierentiable.
8. (*) Study the dierentiability of the function described in p
Math 3610.001 Homework 8
Due 11-1-13
Mance
For this homework you may use any dierentiation formulas you know
from calculus.
1. 26.6
2. 26.10
3. 26.15
4. 26.21
5. 27.4 (a)-(d)
1
Math 3610.001 Homework 10
Due 11-15-13
Mance
For this homework you may use any dierentiation formulas you know
from calculus.
1. Understand example 30.3.
2. 30.5
3. 30.6
4. 30.14
5. 30.18
6. 30.19
7. 30.21
8. 31.4 and 31.5
9. 31.10
10. 31.12
11. 31.17
12.
Math 3610.001 Homework 9
Due 11-8-13
Mance
For this homework you may use any dierentiation formulas you know
from calculus.
1. Let f : (b, ) R and let k R. Prove that limx k/f (x) = 0 whenever
limx f (x) = .
2. Suppose that f (x) = g(1/x) for x > 0 and le
Math 3610.001 Homework 7
Due 10-25-13
Mance
For this homework you may use any dierentiation formulas you know
from calculus.
1. Prove that if a polynomial p(x) is divisible by (x a)2 , then p (x) is divisible
by (x a).
2. Prove that if f is a dierentiable
Math 3610.001 Homework 5
Due October 11th, 2013
Mance
1. Suppose that f : (a, b) R is continuous and that f (r) = 0 for every rational
number r (a, b). Prove that f (x) = 0 for all x (a, b).
2. Let f : R R. Prove that f is continuous on R i f 1 (H) is a c
MATH 3610.005 HOMEWORK 1
(1) Proof. Given x (a, b), let (xn ) be a sequence of rationals in (a, b) converging to x. Since f is continuous at x, f (x) = lim f (xn ) = limn 0 = 0.
(2) Proof. Suppose f is continous on R and let H be a closed set. Then R \ H