MATH 115 - HW4 SOLUTIONS
1. Solutions
These solutions are sketched. You should always give a little more detail.
(1) Let xn be the nth element of the sequence in the problem. Then note that xn+1 =
n+1
MATH 115 - HW3 SOLUTIONS
1. Solutions
These solutions are sketched. You should always give a little more detail.
2
+4a
(1) (Exercise 9.3) In order to show sn := an2 +1n converges we will use the limit
Math 115 HW 8
Theo Vadpey
December 2, 2015
Ross 19.1
Show if the functions are uniformly continuous on their intervals:
Part e)
f (x) = x13 on (0, 1]
Consider the sequence Sn = n1 : n N on (0,1]. It i
HOMEWORK 5
Math 115
Due Friday 11/06/2014 in class
Chapter 3:
20.13
20.16
Chapter 5:
28.3
28.8
28.16
29.13
18.8
19.2
19.4
19.6 (ignore parts involving derivatives)
19.7 a)
SECTIONS COVERED OR EXPECTED TO BE COVERED
Numbers refer to sections not chapters. That is 9 means the 3rd sections of chapter 2. These numbers
correspond to sections in the second edition. Email me i
Math 115, Practice Midterm
Solutions
Problem 1
Suppose that (0, 1] were closed. Then for any converging sequence (xn ) in (0, 1], we should have
limn xn (0, 1]. Take for example xn = 1/n. Then xn 0 an
Solution Keys
Page 1
Note: The homework you turn in must contain each problem statement in its entirety and followed by its solution, as demonstrated in the rst problem below. Others below are sketche
Math 115, HW 2
Solutions
Problem 1
1
. We claim an is irrational for all n and further that
+ 2n
lim an = 0. If an were rational, then n2 + n would be an integer, so n2 + 2n = k 2 for some
(a) For n a
HW7: Math 115 Functions of a real variable
Due on Dec 1st during class
(1)
(2)
(3)
(4)
(5)
(6)
(7)
32.2
32.6
32.7
33.3 (a)
33.4
33.7
Assume that for all n, the function fn (x) is integrable on [a, b],
HW3: Math 115 Functions of a real variable
Due on Oct 20th during class
(1)
(2)
(3)
(4)
10.6
10.12
11.10
Let a > 0. Define sn such that s1 = a and sn+1 = 12 (sn + san ), prove that sn converges and
fi
HOMEWORK 3
Math 115
Due Friday 10/24/2014 in class
Chapter 2:
12.12
13.12 (Prove only for R. Prove straight from the definition of compactness without using the
Heine-Borel Theorem)
Chapter 3:
17.12
1
HOMEWORK 9
Math 115
Due Friday 12/04/2014 in class
Chapter 32:
32.3, 32.7
Chapter 33:
33.7, 33.9, Prove the dominated convergence theorem.
Chapter 37:
37.1
Math 115, Homework 5 Solutions
Problem 1
Let f (x) =
1
. Then f is continuous on S and
x x0
|f (xn )| =
1
,
|xn x0 |
so f is unbounded on S.
Problem 2
This follows directly from the intermediate valu
Math 115, Homework 7 Solutions
Problem 1
Suppose that limx0+ f (x) exists and is equal to L. (the cases in which L is innite are dealt
with similarly). Let
> 0. Then there exists > 0 such that wheneve
Math 115, Homework 6 Solutions
Problem 1
(a) Let
> 0. Then there exists 1 > 0 such that whenever |x a| < 1 , we have |f (x) 3| < /6.
Similarly there is 2 > 0 such that whenever |x a| < 2 , we have |g(
Math 115, Homework 9 Solutions
Problem 1
(a) For any partition P , we have L(g, P ) = 0 so L(g) = 0. For U (g, P ) we let h(x) = x2 for
all x and note that on an interval [tj1 , tj ] we have sup g = s
Math 115, Homework 4
Solutions
Problem 1
(a) The middle inequality is obvious, so well only prove the rst and the third inequality.
For n > N , we have
s1 + . . . sn
s1 + . . . sN
N infcfw_sn | n > N
Math 115, HW 3
Solutions
Problem 1
Let
> 0. Since sn 0, there is some N such that for all n > N , we have |sn | <
|tn | M for all n, it follows that for all n > N , we have |sn tn | < , so sn tn 0.
M
HOMEWORK 1
Math 115
Due Friday 10/3/2014 in class
Chapter 1:
1.2
1.9
2.2
3.8
4.10
4.16
6.6
Let A=0* union cfw_p in Q : p >= 0 and p^2<2. Prove that A is a Dedekind cut and also that cfw_xy : x is in Q
HW1: Math 115 Functions of a real variable
Due on Oct 6th during class
(1) Mathematical induction: 1.3, 1.9
(2) Rational zeros theorem: 2.5
(3) sup and inf: 4.7, 4.12, 4.14, 4.15
1
Math115 Homework 6 solutions
November 30, 2016
(1) (a) We need to use the fact that if h0 = 0 in some open interval, then h = const in this
interval. Since f 00 = (f 0 )0 , we have f 00 (x) = 0 for al
HW6: Math 115 Functions of a real variable
Due on Nov 17th during class
(1)
(2)
(3)
(4)
(5)
(6)
29.7
29.10
29.17
29.18
31.2
31.5 (you can use the fact that limx+ ex xn = 0 for any n N)
1
Math 115:
Homework 4
This homework is worth 20 points. Your grade will based on your 5 best problems. Extra points
(maximum of 5) will be saved for the next homework assignment.
q p
p
1. (2 pts) Sho
MATH 115 - HW6 SOLUTIONS
These solutions are sketched. You should always give a little more detail.
(1) (Exercise 17.4) We need to show that for any sequence xn x in [0, ) that xn
x. As the hint sugg
Math 115:
Homework 5
Do 20 points.
1. (5 pts) Do problem 14.1 of Ross, items (a), (d), (e).
2. (5 pts) Do problem 14.2 of Ross, items (a), (d), (e).
3. (5 pts) Do problem 14.3 of Ross, items (b), (d),