Math 101. Rumbos
Spring 2010
1
Solutions to Exam 1 (Part II)
1. For a subset, , of the real numbers and a real number, , dene the following
sets
(i) + = cfw_ = + where , and
(ii) = cfw_ = where .
Prove the following statements.
(a) If is nonempty and boun
Math 101. Rumbos
Spring 2010
1
Solutions to Exam 1 (Part I)
1. Provide concise answers to the following questions:
(a) A subset, , of the real numbers is said to be bounded if there exists a
positive real number, , such that
for all .
Give the negation
Math 101. Rumbos
Fall 2012
1
Solutions to Exam 1 (Part II)
1. Let be a nonempty subset of . Prove that if is an upper bound for
and , then = sup .
Proof: Assume that is nonempty and that is an upper bound for . Then,
by the completeness axiom, sup() exis
Math 101. Rumbos
Spring 2010
1
Solutions to Exam 2 (Part I)
1. Let ( ) denote a sequence of real numbers.
(a) State precisely what the statement ( ) converges means.
Answer: ( ) converges means that there exists some
such that, for every > 0, there exist
Math 101. Rumbos
Fall 2012
1
Review Problems for Exam #1
1. Let denote a nonempty subset of the real numbers which is bounded below.
Dene
= cfw_ is a lower bound for .
Prove that is nonempty and bounded above, and that sup = inf .
2. Prove that, for any
Math 101. Rumbos
Spring 2010
1
Solutions to Exam 2 (Part II)
1. Let ( ) denote a sequence of real numbers. For a xed , dene
= +
for all ;
that is; 1 = +1 , the ( + 1)th term in the sequence ( ), 2 is the ( + 2)th
term, and so on.
(a) Prove that ( ) conve
Math 101. Rumbos
Fall 2012
1
Solutions to Review Problems for Exam #2
1. Suppose that the sequence ( ) converges to = 0, where = 0 for all .
( )
1
1
Prove that the sequence
converges to .
Proof: Assume lim = , where = 0. Then, there exists 1 such that
1
Math 101. Rumbos
Fall 2012
1
Solutions to Review Problems for Exam #1
1. Let denote a nonempty subset of the real numbers which is bounded below.
Dene
= cfw_ is a lower bound for .
Prove that is nonempty and bounded above, and that sup = inf .
Solution:
Math 101. Rumbos
Spring 2010
1
Solutions to Review Problems for Exam #1
1. Let denote a nonempty subset of the real numbers which is bounded below.
Dene
= cfw_ is a lower bound for .
Prove that is nonempty and bounded above, and that sup = inf .
Solution
Math 101. Rumbos
Fall 2012
1
Review Problems for Exam #2
1. Suppose that the sequence ( ) converges to = 0, where = 0 for all .
( )
1
1
Prove that the sequence
converges to .
2. Let ( ) denote a sequence that converges to . Prove that for any ,
lim = .
3.
Math 101. Rumbos
Fall 2012
1
Handout #1: Mathematical Reasoning
1
Propositional Logic
A proposition is a mathematical statement that it is either true or false; that is, a
statement whose certainty or falsity can be ascertained; we call this the truth val
Math 101. Rumbos
Spring 2010
1
Solutions to Review Problems for Exam #2
1. Suppose that the sequence ( ) converges to = 0, where = 0 for all .
( )
1
1
Prove that the sequence
converges to .
Proof: Assume lim = , where = 0. Then, there exists 1 such that
Math 101. Rumbos
Fall 2012
1
Solutions to Exam 1 (Part I)
1. Provide concise answers to the following questions:
(a) Give the negation of the following statement:
For every > 0, there exists such that
< .
Answer: There exists > 0 such that, for all there
Math 101. Rumbos
Fall 2012
1
Exam 1 (Part II)
Friday, October 12, 2012
Name:
This is the outofclass portion of Exam 1. There is no time limit for working on the
following three problems. You are only allowed to consult Handout #2 on the axioms
of the real
Math 101. Rumbos
Fall 2012
1
Exam 1 (Part I)
Friday, October 12, 2012
Name:
Provide complete arguments when asked to prove a statement in a question. You will
be graded on how well you organize your proofs as well as the logical ow or your
deductions. In
Math 101. Rumbos
Fall 2012
1
Handout #2: The Real Numbers System Axioms
I. Field Axioms
The set of real numbers has two algebraic operations: addition (the sum of
any two elements and of being denoted by + ) and multiplication
(the product of any two elem