Economics 204
Fall 2012
Problem Set 1 Suggested Solutions
1. Use induction to prove the following statements.
(a) The equality
(b) The inequality
n
3
i=1 i =
n
1
i=1 i
(
n
i=1
2
i) holds for all n N;
Economics 204 Lecture 11Monday, August 10, 2009 Revised 8/10/09, Revisions indicated by * and Sticky Notes Sections 4.1-4.3, Unified Treatment Definition 1 Let f : I R, where I R is an open interval.
Economics 204 Lecture Notes on Measure and Probability Theory This is a slightly updated version of the Lecture Notes used in 204 in the summer of 2002. The measure-theoretic foundations for probabili
Economics 204 Lecture 11Monday, August 10, 2009 Sections 4.1-4.3, Unified Treatment
Definition 1 Let f : I R, where I R is an open interval. f is differentiable at x I if f(x + h) - f(x) =a h0 h lim f
University of California, Berkeley Department of Economics Econ 204 Mathematical Tools for Economists Summer/Fall 2009 Revised 7/24/09, reordering the material in Lectures 11 and 12 (see * below). Ins
Economics 204 Summer/Fall 2008 Lecture 1Monday July 27, 2009 Bob Anderson Vladimir Asriyan Hui Zheng Website: http:/emlab.berkeley.edu/users/anderson/Econ204 /204index.html Lectures will often run pas
Economics 204 Lecture 2, July 28, 2009 Section 1.4, Cardinality (Cont.) Theorem 1 (Cantor) 2N , the set of all subsets of N, is not countable. Proof: Suppose 2N is countable. Then there is a bijection
Economics 204 Lecture 3Wednesday, July 29, 2009 Revised 7/29/09, Revisions Indicated by * and Sticky Notes Section 2.1, Metric Spaces and Normed Spaces Generalization of distance notion in Rn Definiti
Economics 204 Lecture 4Thursday, July 30, 2009 Revised 7/31/09, Revisions Indicated by * and Sticky Notes Section 2.4, Open and Closed Sets Definition 1 Let (X, d) be a metric space. A set A X is open
Economics 204 Lecture 5Friday, July 31, 2009 Section 2.6 (Continued) Properties of Real Functions
Theorem 1 (6.23, Extreme Value Theorem) Let f be a continuous real-valued function on [a, b]. Then f a
Economics 204 Lecture 6Monday, August 3, 2009 Revised 8/4/09, Revisions indicated by * and Sticky Notes Section 2.8, Compactness Definition 1 A collection of sets U = cfw_U : in a metric space (X, d)
Economics 204 Lecture 7Tuesday, August 4, 2009 Revised 8/5/09, Revisions indicated by * and Sticky Notes Note: In this set of lecture notes, A refers to the closure of A. Section 2.9, Connected Sets D
Economics 204 Lecture 8Wednesday, August 5, 2009 Revised 8/5/09, Revisions indicated by * and Sticky Notes Chapter 3, Linear Algebra Section 3.1, Bases Definition 1 Let X be a vector space over a fiel
Economics 204 Lecture 9Thursday, August 6, 2009 Revised 8/6/09, revisions indicated by * and Sticky Notes Section 3.3 Supplement, Quotient Vector Spaces (not in de la Fuente): Definition 1 Given a vec
Economics 204 Lecture 10Friday, August 7, 2009 Revised 8/8/09, Revisions indicated by * and Sticky Notes Diagonalization of Symmetric Real Matrices (from Handout): 1 if i = j 0 if i = j A basis V = cf
Economics 204 Lecture 12Tuesday, August 11, 2009 Revised 8/12/09, Revisions indicated by * and Sticky Notes Inverse and Implicit Function Theorems, and Generic Methods: Section 4.3 (Conclusion), Regul
Economics 204 Lecture 13Wednesday, August 12, 2009 Revised 8/12/09, revisions indicated by * and Sticky Notes Section 5.5 (Cont.) Transversality Theorem The Transversality Theorem is a particularly co
Economics 204 Summer/Fall 2008 Lecture 1Monday July 27, 2009 Bob Anderson Vladimir Asriyan Hui Zheng Website: http:/emlab.berkeley.edu/users/anderson/Econ204 /204index.html Lectures will often run pas
Economics 204 Lecture 2, July 28, 2009 Section 1.4, Cardinality (Cont.) Theorem 1 (Cantor) 2N , the set of all subsets of N, is not countable. Proof: Suppose 2N is countable. Then there is a bijection
Economics 204 Lecture 10Friday, August 7, 2009 Revised 8/8/09, Revisions indicated by * and Sticky Notes Diagonalization of Symmetric Real Matrices (from Handout): 1 if i = j 0 if i = j A basis V = cf
Economics 204 Lecture 10Friday, August 7, 2009
Diagonalization of Symmetric Real Matrices (from Handout): Definition 1 Let ij =
1 if i = j 0 if i = j
A basis V = cfw_v1 , . . . , vn of Rn is orthon
Economics 204 Lecture 9Thursday, August 6, 2009 Revised 8/6/09, revisions indicated by * and Sticky Notes Section 3.3 Supplement, Quotient Vector Spaces (not in de la Fuente): Definition 1 Given a vec
Economics 204 Lecture 9Thursday, August 6, 2009 Section 3.3 Supplement, Quotient Vector Spaces (not in de la Fuente): Definition 1 Given a vector space X and a vector subspace W of X, define an equiva
Economics 204 Lecture 8Wednesday, August 5, 2009 Revised 8/5/09, Revisions indicated by * and Sticky Notes Chapter 3, Linear Algebra Section 3.1, Bases Definition 1 Let X be a vector space over a fiel
Economics 204 Lecture 8Wednesday, August 5, 2009 Chapter 3, Linear Algebra Section 3.1, Bases
Definition 1 Let X be a vector space over a field F . A linear combination of x1, . . . , xn is a vector o
Economics 204 Lecture 7Tuesday, August 4, 2009 Revised 8/5/09, Revisions indicated by * and Sticky Notes Note: In this set of lecture notes, A refers to the closure of A. Section 2.9, Connected Sets D
Economics 204 Lecture 7Tuesday, August 4, 2009 Note: In this set of lecture notes, A refers to the closure of A. Section 2.9, Connected Sets Definition 1 Two sets A, B in a metric space are separated
Economics 204 Lecture 6Monday, August 3, 2009 Revised 8/4/09, Revisions indicated by * and Sticky Notes Section 2.8, Compactness Definition 1 A collection of sets U = cfw_U : in a metric space (X, d)
Economics 204 Lecture 6Monday, August 3, 2009 Section 2.8, Compactness Definition 1 A collection of sets U = cfw_U : in a metric space (X, d) is an open cover of A if U is open for all and U A ( may
Economics 204 Lecture 5Friday, July 31, 2009 Section 2.6 (Continued) Properties of Real Functions
Theorem 1 (6.23, Extreme Value Theorem) Let f be a continuous real-valued function on [a, b]. Then f a
his paper inspects the impact of stuns saw in the Stock Market on yield and vocation. Amid the Great
Depression. We display three principle discoveries. Initially, an unfavorable budgetary stun prompt
he chapter shall looks at the background of the study, statement of the problem, research questions,
objectives of the study, purpose of the study, hypothesis, assumptions of the study, delimitation o