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Economic Analysis I

by: Alfredo Keebler

Economic Analysis I 06E 200

Marketplace > University of Iowa > Economcs > 06E 200 > Economic Analysis I
Alfredo Keebler
GPA 3.81


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This 13 page Class Notes was uploaded by Alfredo Keebler on Friday October 23, 2015. The Class Notes belongs to 06E 200 at University of Iowa taught by Staff in Fall. Since its upload, it has received 60 views. For similar materials see /class/228102/06e-200-university-of-iowa in Economcs at University of Iowa.


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Date Created: 10/23/15
KEY CONCEPTS Introduction to Real Analysis Samvel Atayan and Brent Hickman Summer 2008 1 Sets and Functions PRELIMINARY NOTE Many de nitions given in these notes are framed in terms speci c to the real numbers This simpli es matters greatly because of the familiar ordering and distance concepts which come as standard fea tures in nite dimensional Euclidean space However7 many ofthe concepts given below have useful analogs in more exotic spaces eg7 spaces of func tions or spaces of in nite sequences7 but those require some knowledge of metric topology7 which will be a major emphasis of your Economic Analysis I 06E200 class For the purpose of math camp7 we will resign ourselves to the quaint shackles of Euclidean space 11 Sets De nition 1 A set is a collection of elements o If an element x is in a set A then we denote z 6 A7 if z is not in A7 then we write z A o If every element of a set A is also in set B7 then A is a subset of B and we denote this as A Q B or B Q A o A is a proper subset of B if there is at least one element in B that is not in A We denote this by A C B or B D A 0 Two sets are equal if they contain all the same elements A B That is7 A Q B and B Q A o A set is de ned by listing its elements or by specifying the property that determines the elements of the set x E A 137 where P is some property 0 Examples 12 gt Natural numbers N 17 2737 gt Integers Z 07 17 717 27 727 gt Rational numbers Q m7n E Z and n 31 0 gt Real numbers R gt S 2k k e N Set operations 0 The union of sets A and B is the set AUBxz Aorx B o The intersection of the sets A and B is the set A Bxz Aand B o The complement of B relative to A or the difference between A and B is the set AB Aandx B o A set with no elements is called the empty set and is denoted by Q 0 Two sets are disjoint if they have no elements in common A B Q o DeMorgan laws 13 gt union Uf1An x z E An for some n E N gt intersection leAn x z E An for all n E N Cartesian product of two sets is a set of all ordered pairs AgtltBa7ba A7b B A ball of radius 8 around z in R is a set de ned by Bgz y E R l96 7 ill lt 8 For a set A7 a limit point of A is a point l E R such that for each 8 gt 07 there exists some y E A such that y E ESQ Functions and Mappings A function f from a set A into a set B7 denoted f A a B7 is a set f of ordered pairs in A gtlt B such that for each x E A there exist a unique b E B with Lab 6 f The set A of rst elements of function f is called the domain of f and is denoted Df The set E of all second elements in f is called the range of f and is denoted by Rf Note that Df A7 while Rf Q B For the following7 consider a function f7 mapping A into B Let E Q A7 then the direct image ofE under f is the subset fE Q B given by fE f96 96 E E then the inverse image of H under f is the subset Let H g B f H Q A given by 1 f 1H 96 E A f96 E H o The function f is injective oneto one if whenever 1 31 x2 then f961 3quot f962 The function f is surjective onto if fA B That is7 Rf B If f is both injective and surjective then it is bijective o If f A a B is a bijection of A onto B7 then f 1 b7a e B gtlt A Lab 6 f is a function mapping B into A7 called the inverse of f gt IMPORTANT NOTE inverse images always exist inverse func tions exist ONLY for bijections o lffAaBandgB C7andifRf Dg B7thenthe composite function g o f is the function from A into 0 de ned by 9 o fz gfz for all z e A Theorem 1 Let f A a B and g B a C be functions and let H Q 0 Then 9 0 f 1H f 19 1H 14 Finite in nite and countable sets 0 The empty set Q has zero elements o If n 6 N7 a set A has n elements if there exist a bijection from Nn 17 onto A o A set A is nite if it either empty or it has n elements for some n E N o A set A is in nite if it is not nite Theorem 2 Suppose that S and T are sets and that T Q S 1 ifS is a nz39te set then T is a nz39te set 2 IfT is an in nite set then S is an in nite set o A set S is denumerable countably in nite if there exists a bijec tion of N onto S o A set S is countable if it is either nite or denumerable o A set S is uncountable if it is not countable Theorem 3 Suppose that S and T are sets and that T Q S 1 ifS is a countable set then T is a countable set 2 IfT is an uncountable set then S is an uncountable set Exercise 1 Prove that Z is a countable set Exercise 2 Prove that Q is a countable set Exercise 3 Prove that RQ is an uncountable set HINT proceed by contra diction and use the fact that the irrationals are expressed by noniterminating nonirepeating decimals with comparisons being performed in lepicographic fashion Theorem 4 The countable union of t L sets is t L is o A set S Q T is said to be dense in T if for any two members of T say z and y there epists s E S such that z lt s lt y Exercise 4 Prove that Q is dense in itself Exercise 5 Prove that Q is dense in R HINT use the familiar lepicographic representation of real numbers Exercise 6 Prove that R Q is dense in R HINT use the familiar lepico graphic representation of real numbers 2 The completeness property of R Now we consider the set of real numbers 0 LetSQR andSy Q Then 1 The set S is bounded above if there eccist a numberu E R such that s S u for all s E S Then each such u is an upper bound of S The set S is bounded below if there CCElStS a number w E R such that w 3 s for all s E S Each such w is called a lower bound of S 3 A set S is bounded if it is both bounded above and below A set is unbounded if it is not bounded 16 o LetSQRandSy Q If S is bounded above then a number u is a supremum or a least upper bound lub ofS if 1 u is an upper bound of S7 and 2 ifi is any upper bound of S7 then u S 1 US is bounded below then a number w is an in mum or a greatest lower bound glb if 1 w is a lower bound of S and 2 ifr is any lower bound of S then r S w Note that there is only one supremum of a given set S Q R The set S should have upper lower bound in order to have the supremum in mum It does not have to be an element of the set S Also it is unique Notation sup S and infS Lemma 1 An upper bound u of a nonempty set S Q R is the supremum of S if and only iffor every 8 gt 0 there eccist an 55 E S such that u 7 8 lt 55 Theorem 5 The Completeness Property of R S39upremum Property of R Every nonempty set of real numbers that has an upper bound also has a supremum in R Consider the range of the function f D a R 0 Then f is bounded above if the set fD z E D is bounded above in R That is there eccz39st B E R such that u S B for all z E D 0 Then f is bounded below if the set fD z E D is bounded below in R 0 Then f is bounded ifz39t is bounded above and below That is 3B 6 R such that S B for Vd E D Some properties 0 Iffz S gd for allz E D then sup f 96 S SEE 996 16D 0 Iffz S gy for all Ly E D then lt f 323fm 7 323 21 The Archimedean Property Theorem 6 Archimedean Property fr 6 R then there eccz39st n1 6 N such that z lt n1 Corollary 1 IfS l n E N then infS 0 Corollary 2 ft gt 0 there edists nt 6 N such that 0 lt 7 lt t 3 Limits of functions De nition 2 Suppose f A a R Let x a p andp is a limit point of A Then the limit of f as x appmaches p is de ned as limmaz7 f q if such point q epists and for every 8 gt 0 there eccist 6 gt 0 such that if pizi lt6 then f7q lt8 Theorem 7 Suppose fg A a R and limmaz7 f q and limwapgw r Then 1 limwp W 9 q r 2 limmapfgz qr 3 limmaz7 ifr 31 0 Theorem 8 L Hopital s rules Let 700 S a lt b S 00 and functions f and g are di erentiable on ab such that g z 31 0 for all z 6 ab Suppose that limmw f 0 limmw g z a if hmH L e R then hmH L b ifhmH L e 700 00 then hmH L Suppose that limmw f ioo limmw g z a if hmH L e R then hmH L b iflimmw L E 700 00 then limmw L Exercise 7 CRRA Preferences use L Hopital s rules to show that prefer ences given by 01 7 7 y 1 7 v approach log preferences as y a 1 4 Sequences o A sequence of real numbers is a function f N a R Therefore the sequence can be denoted by f1f2 Usually we denote the sequence by en1 where xn 112 i 2 3 n 1 7171771717717quot39 bn bforVn E N 41 The Limit of a Sequence 0 A sequence of real numbers is said to converge to point z E R or z is a limit of iffor every 8 gt 0 there eccist Ng E N such that for all n 2 Ng we have id 7 cl lt e If sequence has a limit then it is convergent if it has no limit then it is divergent Notation lim znd orznaz Hoe Theorem 9 Let be a sequence in R and let x E R Then con verges to x if and only iffor any 8 gt 0 Bgz about z ecccludes at most a nite number of elements of Exercise 8 Prove that a sequence can converge to at most one point Example 1 a lim 0 b 0i c hm 3 o In order to show that a sequence does not converge to x we have to nd one number 8 gt 0 such that for any N E N one can nd a particular n 2 N such that law 7 cl 2 6 Alternatively we need only nd one 8 gt 0 such that Bgz ecccludes in nitely many members of the sequence Note however that showing that diverges is not as simple NH 42 Limit Theorems 0 A sequence in R is bounded if there eists M E R and M gt 0 such that S M for all n E N Theorem 10 A convergent sequence in R is bounded Theorem 11 Let and be two sequences in R such that xn a z and yn a y Then the sequences on yn7 zn 7 yn 7 znyn7 and czn converge to z y7 z 7 y7 zy7 C7 respectively Theorem 12 Ifxn a x 2 a 2 2n 31 0 for all n E N and z 31 0 then the quotient sequence a Theorem 13 Ifxn and are convergent sequences in R and ifxn S yn for all n 6 N7 then limiH00 xn S limiH00 y Theorem 14 If is a convergent sequence in R and ifa S xn S b for all n 6 N7 then a S limiH00 xn S b Theorem 15 Let be sequences in R If xn lt yn lt 2 for all n E N and limiH00 xn limiH00 2 then is convergent with limiH00 xn limiH00 yn limiH00 2 Exercise 9 Prove the above theorem Theorem 16 foil a 7 then a Exercise 10 Prove the above theorem Theorem 17 Let xn a z and xn 2 0 Then sequence a Exercise 11 Prove the above theorem Theorem 18 Let xn be a sequence of positive real numbers such that L 1 eccists IfL lt 1 then converges and xn a 0 limnaoo m Theorem 19 Let xn be a sequence of real numbers converging to x Then any rearrangement of the members of the sequence will also converge to x Exercise 12 Prove the above theorem 10 43 Monotone Sequences o A sequence is nonidecreasing if 1 3 2 3 S xn S xn S If the inequalities are strict we say it is increasing A sequence xn is non4increasing if 1 2 2 2 2 xn 2 xn 2 If the inequalities are strict we say it is decreasing Sequence is monotone if it is either noniincreasing or nonidecreasing Theorem 20 Monotone Convergence Theorem A monotone sequence of real numbers is convergent if and only if it is bounded 0 Let be a sequence of real numbers and let n1 lt n2 lt lt nk lt be a strictly increasing sequence of natural numbers Then the sequence xnk xnmxnm quotan is called a subsequence of Theorem 21 If xn a 00 then every subsequence znk a x as well Theorem 22 Monotone Subsequence Theorem If is a sequence in R7 then it has a monotone subsequence Theorem 23 Bolzano Weierstrass Theorem Every bounded sequence in R has a convergent subsequence Theorem 24 If every subsequence of converges to z E R then converges to x o A sequence in R is a Cauchy sequence iffor every 8 gt 0 there eccist NE E N such that for all n7m 2 NS7 n 7 mg lt 8 Theorem 25 If is convergent then is Cauchy Theorem 26 A Cauchy sequence of real numbers is bounded Theorem 27 Cauchy Convergence Criterion A sequence in R is convergent if and only if it is a Cauchy sequence Example 2 Claim is a Cauchy sequence if xn iVn E N Proof Given 8 gt 0 let NE gt Let n7m be greater than NS and WLOG let n lt m Thus we have the following inequalities which prove the result 1 1 1 1 1 on xm nm lt67 where the rst inequality follows from the fact that n lt m D 11 5 Series 0 Consider an in nite series with the general term denoted x the series then has the form 00 Z en n1 0 Let 5 be the n th partial sum given by 2 xi IflimH00 sk epists then series is said to converge Otherwise we say that it diverges o The following condition is necessary for the convergence of a series 2201 Tnquot hm xn 0 Hoe Otherwise the sequence of partial sums will diverge Note however that it is not a su cient condition For emample Consider the series 221 1 sn1i n 1 1 i i Z i V W W The n th partial sum increases without bound as n increases Hence the the series diverges o If is a sequence in R then the series 2 is absolutely con vergent if the series 2 is convergent in R The series is condi tionally convergent if it is buyout151170 but not L t 1 convergent Theorem 28 If a series in R is absolutely convergent then it is convergent Some convergence tests Theorem 29 The nith Term Test If the series 2 converges then hm xn 0 Theorem 30 Cauchy Criterion for series The series 2 converges if and only iffor every 8 gt 0 there epists NE E N such that ifm gt n 2 NS then 15m 7 51 n1 xn2 m1lt 8 12 Theorem 31 Comparison Test Let and be real sequences such that for some K E N 0 zn ynforalln2K 1 Then the convergence onyn implies the convergence onzn 2 The divergence onzn implies the divergence onyn Theorem 32 Root Test Given me let oz lirruH00 7 an Then if 04 lt 17 Zen converges ifa gt 1 then 2 diverges and ifa 17 then the test provides no information Theorem 33 Ratio Test The series Zen converges if wn1 mn lt 17 and wn1 1n diverges if 2 1 0 Given a sequence en in R7 the series chx is called the power series The numbers cn are called the ooe lcients of the series Let oz lirruH00 The radius of convergence is given by R 3 Then Zena are convergent if lt R7 and diverges if gt R


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