MATH 131A HOMEWORK 5
(1) Let (an ) and (bn ) be sequences and suppose that there exists N N such that an bn
for all n N . Prove that lim sup an lim sup bn .
(2) Let (an ) and (bn ) be sequences.
(a) S
MATH 131A HOMEWORK 3
(1) Suppose that (an ) is a convergent sequence and for each n 1 let bn = an+1 . Prove
that (bn ) is also convergent and that
lim bn = lim an .
n
n
(2) Let (an ) be the sequence r
MATH 131A HOMEWORK 2
(1) Prove Lemma 4.2 from the course notes: For all x Q we have
|x| x |x|.
(2) Prove Lemma 4.3 from the course notes: For all x, y Q we have
|x| |y|
if and only if
|y| x |y|.
Note
MATH 131A HOMEWORK 6
(1) Suppose that (ak ) is a sequence with the property that ak cfw_0, 1, 2, . . . , 9 for all k 1.
Prove that
X
ak
10k
k=1
is a convergent series. Write a brief sentence about why
MATH 131A HOMEWORK 4
(1) Suppose that S R is bounded. Prove that inf S = sup(S) (here S means the
set of all real numbers s where s S). Hint: let s0 = sup(S). The definition of inf
requires two things
MATH 131A HOMEWORK 1
(0) (nothing to turn in for this exercise) Read the first three sections (through p. 15) of the
following handout on writing mathematical proofs (clickable link):
https:/math.berk
MATH 131A: REAL ANALYSIS (BIG IDEAS)
Theorem 1 (The Triangle Inequality). For all x, y R we have
|x + y| |x| + |y|.
Proposition 2 (The Archimedean property). For each x R there exists an n N such
that
MATH 131A: REAL ANALYSIS
NICKOLAS ANDERSEN
The textbook for the course is Ross, Elementary Analysis [2], but in these notes I have
also borrowed from Tao, Analysis I [3], and Abbott, Understanding Ana
Math 131A: Homework 5
Please turn this homework in, to me, at the start of lecture on March 10th. Remember, the
quiz will on the same material, so not doing this homework would be very silly. The quiz
Math 131A: Homework 4
Please turn this homework in, to me, at the start of lecture on February 24th. Remember, the
quiz will on the same material, so not doing this homework would be very silly. The q
Math 131A: Homework 2
Please turn this homework in, to me, at the start of lecture on January 27th. Remember, the
quiz will on the same material, so not doing this homework would be very silly. The qu
Math 131A: Homework 3
Please turn this homework in, to me, at the start of lecture on February 10th. Remember, the
quiz will on the same material, so not doing this homework would be very silly. The q
Math 131A: Preparation Questions for Quiz 1
1. (a) Prove the following sentence is true.
2
2
a R \ cfw_0, b R, c R, b 4ac 0 = x R : ax + bx + c = 0 .
(b) Prove the following sentence is true. (Id save
Math 131A, Fall 2016
Practice Midterm
Version 1
Last name
First name
Student ID
Use the provided space for your solutions (you may use the additional page at the end). Show your work. Write only one s
131A Homework 4- Due Friday, October 21.
Chapter 1 # 11.1,11.2,11.4,11.6 And the following exercises:
1. Suppose (an )
n=1 is a sequence and that an cfw_0, 1, . . . , 9 for each n N.
P
k
Prove that th
HOMEWORK 3: SOLUTIONS/HINTS
8.4 Let > 0, then there is an N such that |sn | < M for n N . So we get that |sn tn | M |sn | < for all n N and hence lim(sn tn ) = 0. 8.5 Let > 0, then we have N N such th
HOMEWORK 4: SOLUTIONS/HINTS 10.1 (d) Let sn := sin( n ), then |sn | 1, hence the sequence is bounded. 7 However, it is neither increasing nor decreasing. (f) Let an := 3n . Then an > 0 for all n, so t
131A Homework 3- Due Friday, October 14.
Chapter 1 # 9.1,9.4,9.9,9.12,9.15,
10.1,10.6,10.10,10.12
And the following exercise:
Suppose (sn )
n=1 is a sequence such that for each k N with k 2, the subse
131A Homework 1- Due Friday, September 30.
Chapter 1 # 1.1,1.3,1.6,1.10,1.11,4.1,4.7.4.10
And the following exercice:
For each n N, let Pn denote the assertion n2 + 5n + 1 is an odd integer.
1. Prove
Analysis, Math 115A, Fall 2016 - Course Info
Instructor: Martin Tassy, [email protected],
Instructor Office Hours: WTh 9-10am in 6324 Science Building
Lectures: Monday, Wednesday and Thursday, 11am
Math 115A, Fall 2016
Practice Midterm
Version 1
Last name
First name
Student ID
Use the provided space for your solutions (you may use the additional page at the end). Show your work. Write only one s
131A Homework 5- Due Friday, November 3.
Chapter 3 # 17.1,17.2,17.3,17.417.7,17.8,17.11,17.15 And the following exercises:
1. It is true that limx3 (x3 + 1) = 28.
(a) Verify the sequence definition us