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 appro
Homework 3
Math 25
April 22, 2011
1. Prove that the irrational numbers are dense in R. Hint: use denseness of rational numbers and the fact that 2 is
irrational (no proof is needed for this fact).
Pro
Math 25 Practice Midterm 2 Solutions
Andrew Herrmann
May 29, 2012
1. State the monotone convergence theorem.
In book.
2. Let x1 5. Define inducitvely the sequence (xn ) as xn+1 =
its limit.
5xn . Show
MAT-25. First Midterm Test (Sample).
April 19, 2012
All problems should be treated as proofs (unless it involves only
stating a theorem, an axiom or a definition, or if the problem states
no proof is
MAT-25. Homework 9.
Due: June 5
1. Suppose that (xn ), (yn ) are sequences. Show that
supcfw_xn + yn : n N supcfw_xn : n N + supcfw_yn : n N
2. Use problem 1 to show that
lim sup(xn + yn ) lim sup
MAT-25. Homework 5
Due on May 8, 2012
1. Show that union of two countable sets is again countable.
2. (a) Give an example of a sequence of irrational numbers converging to a rational numbers.
(b) Show
MAT-25. Homework 2
Due on April 17, 2012
Recall that natural numbers are the numbers 1, 2, 3, ., they are
obtained by adding 1 R to itself certain number of times.
1. Which of the properties A1-A4, M1
Math 25 Homework 8 Solutions
Jason Barnett
May 29, 2012
1. Claim: Any enumeration rn of Q contains a subsequence rnk which diverges to .
Proof. Let rn be an enumeration of Q. Now choose n1 , n2 , . .
MAT-25. Homework 4
Due May 1.
1. Prove that |b| < a if and only if a < b < a.
2. Show that |a b| < c if and only if b c < a < b + c.
3. Use induction to prove
|a1 + . + an | |a1 | + . + |an |
for n re
MAT-25. Homework 1: Due April 10.
April 3, 2012
1. Do Problem A.2.8 (a, b) from the textbook.
2. (a) Prove that sum of rational and irrational number is always
irrational (Problem A.6.1).
(b) What abo
MATH-25. Handout: lim sup and lim inf
May 23, 2012
This is the handout for section 2.13 covered in class on May 23.
First, some conventions:
1. < x < for every x R. We will define the set of extended
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 ques
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.
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
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
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 fo
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.
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 wi
MAT-25. Floor function
Michael Kapovich
April 16.
Theorem. (Existence of the floor) For every real number x, there
exists an integer m so that m x < m + 1. The integer m with these
properties is uniqu
MAT-25. Homework 3
Due April 24.
1. Prove that irrational numbers are dense
in R. Hint: Use denseness of rational numbers and the fact that 2 is irrational (no proof is
needed for this fact).
2. (a)
First letter of your last name:
Name:
Section number:
MAT-25. 2nd Midterm Test: Sample.
The exam is closed book, closed notes. Show all the work: Correct answer
without proper justification can result