Problem Set 2, Solutions
Due Friday 8/19
Exercise 2.2.2. Verify, using the definition of convergence of a sequence, that the following
sequences converge to the proposed limit.
Solution (General approach). Note that, as indicated in the problem statement,
Math 25 Homework 1 Solutions
Roger Tian
January 26, 2017
Exercise 1.2.1
2
(a) Proof by contradiction. Assume there exist integers p, q N such that 3 = pq2 , and p, q have
no common factors. It follows that 3q 2 = p2 , so 3 divides p2 . Since 3 is prime, 3
Math 25 Homework 3 Solutions
February 8, 2017
Exercise 1.4.1
If a < 0 < b, then the we automatically get the conclusion of Theorem 1.4.3; just take r = 0.
Now suppose that a < b 0. Then a > b 0. By Theorem 1.4.3, there exists r Q such
that b < r < a. It f
Math 25 Homework 4 Solutions
February 14, 2017
Exercise 2.2.2
This definition can be rewritten as A sequence (xn ) verconges to x if there exists > 0 such
that for all N N we have |xN x| < .
This definition states that any term xn of a vercongent sequence
Math 25 Homework 5 Solutions
February 19, 2017
Exercise 2.3.1
Denote the given sequence by (an ). Let > 0. For all n N we have |an a| = |a a| = 0 < .
This shows that (an ) converges to a.
Exercise 2.3.3
Let > 0. Choose N1 N such that |xn l| < whenever n N
Math 25, Fall 2014.
Nov. 21, 2014.
MIDTERM EXAM 2
KEY
NAME(print in CAPITAL letters, ﬁrst name ﬁrst): _ _-
NAME(sign): _ _
ID#: _ _
Instructions: Each of the 4 problems has equal worth. Read each question carefully and answer it
in the space provided. You
MIDTERM EXAM II
Math 25
Temple-F06
Write solutions on the paper provided. Put your name
on this exam sheet, and staple it to the front of your
nished exam. Do Not Write On This Exam Sheet.
Problem 1. (a) (4pts) Give the precise denition of an upper
bound
Math 25, Fall 2014.
Dec. 16, 2014.
FINAL EXAM
NAME(sign): _ _
ID#: _ _
Instructions: Each of the 8 problems has equal worth. Read each question carefully and answer it
in the space provided. You must Show all your work for full credit. Carefully prove eac
MAT 25B Final Exam (2013/12/12)
Honor Pledge: I pledge on my honor that I have not given or received any unauthorized assistance
on this exam.
Name:
Signature:
1. Show all your work. Jumping to right answers without minimum reasoning deserves no credit.
2
Math 25, Fall 2014.
Oct? 31, 2014.
MIDTERM EXAM 1
Instructions: Each of the 5 problems has equal worth. Read each question carefully and answer it
in the space provided. You must Show all your work for full credit. Carefully prove each assertion you
make
MIDTERM EXAM I-SOLUTIONS Math 25 Temple-F06 Write solutions on the paper provided. Put your name on this exam sheet, and staple it to the front of your finished exam. Do Not Write On This Exam Sheet. Problem 1. (20pts) Use the field axioms for the real nu
FINAL TEST - 25
Name and ID number:
Do not turn this page until instructed to do so
Instructions: Read carefully every problem. Show your work on every
problem. Correct answers with no support work will not receive full credit.
Be organized and use notati
Math 25 Homework 6 Solutions
February 28, 2017
Exercise 2.4.2
(a) We first prove that 0 < xn 3 and xn xn+1 for all n N, by induction on n; in particular,
this will show that (xn ) is bounded and monotone, so that (xn ) converges by the Monotone
Convergenc
Math 25 Homework 7 Solutions
March 7, 2017
Exercise 2.5.1
Let (an ) be a sequence that converges to L R. Suppose (ank ) is a subsequence of (an ). Let
> 0. There exists N N such that |an L| < whenever n N . Pick any j N such that
nj N . Then for all k j
Math 25, Fall 2014.
HW 1 Solutions (mostly adapted from Abbotts Instructors Manual)
1.2.1. (a) We prove this by contradiction. Assume that there exist integers p and q satisfying
2
p
= 3.
q
We may assume that p and q have no common factor. From the above
Homework 1 (Optional)
Due: in class on Wednesday 8/3
Exercise 1. Determine
the remaining truth
from the end of class on
h
i values
h for the proposition
i
Tuesday 8/2, (A = B) & (A = C) = B C , by completing the abbreviated
truth table below and filling i
Problem Sheet 9
1) Show that if
an and
n=1
bn both converge,
n=1
cn = a1 b1 + a2 b2 + a3 b3 + a4 b4 + converges.
then
n=1
2) a) Show that
sin n diverges using the Divergence Test.
n=1
b) Show that lim sin n does not exist, using the facts that
n
1
sin x f
Problem Sheet 8
1) Prove or disprove the following statements:
a) If
an and
n=1
b) If
n=1
an and
n=1
bn are positive-term series and
an diverges, then
n=1
(an + bn ) diverges.
n=1
bn diverge, then
n=1
(an + bn ) diverges.
n=1
2) Prove that absolute conver