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# ADVANCEDCALCULUS MATH360

Penn

GPA 3.56

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This 108 page Class Notes was uploaded by Claudine Friesen on Monday September 28, 2015. The Class Notes belongs to MATH360 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 72 views. For similar materials see /class/215400/math360-university-of-pennsylvania in Mathematics (M) at University of Pennsylvania.

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Date Created: 09/28/15

Number Systems on Basic Number Systems The most first numbers every considered were the whole numbers 123 7 Number Systems 0mg Basic Number Systems The most first numbers every considered were the whole numbers 127 3 7 Then someone realized that it was important to include a number representing nothing This then gave us the natural numbers N123 Number Systems 0mg Basic Number Systems Then people noticed that addition worked better if there were negative numbers This led us to the integers Z7372710123 Number Systems 0mg Basic Number Systems Then people noticed that addition worked better if there were negative numbers This led us to the integers Z7372710123 After dealing with the integers for a while people began to notice the usefulness of fractions and the rational numbers were born Qab Z Number Systems 0mg Square Root of Two What is next 7 Number Systems or s Square Root of Two What is next 7 There is no rational number p such that p2 2 ms Ordered 5939s I A set is a collection of objects If a set has at least one element we say it is non empty If a set has no objects we say it is empty efi n itidn Suppose A is a set 0 Ifx is a member of A we write X E A o Ifx is not a member of A we write X A aDefi39Iii ti o n Suppose A and B are sets a If every element of A is an element of B we say A is a subset ofBandwriteAQBorBQA o If A is a subset of B and not equal to B we say A is a proper subset of B o If A Q B and B Q A then we say the sets are equal and write A B that the following two properties hold i lfx E 5 and y E 5 then one and only one of the following is true 0Xlty oxy yltx ii Forallxyz ifxltyandyltzthenxltz Defj n ition An ordered set is a set 5 in which an order is defined Definition If 5 is an ordered set with lt and X lt y we often say X is less than y WE also often use y gt X in place of X lt y when convenient We will use X g y as a shorthand for X lt y or X y ie X g y if and only if NOT y lt X Numb n 9139 IS Ordered 5939s F39 i isk Examples of Ordered Sets Here are some examples 0 The one point set with nothing satisfying lt Numb n 9139 IS Ordered 5939s F39 i isk Examples of Ordered Sets Here are some examples 0 The one point set with nothing satisfying lt 0 Z with the order a lt b if and only if b7 a is positive Numb n er IS Ordered 5939s F39 i isk Examples of Ordered Sets Here are some examples 0 The one point set with nothing satisfying lt 0 Z with the order a lt b if and only if b7 a is positive 0 Z with the order a lt b if either lal lt lbl o lal a is negative and b is positive Notice that if 5 is an ordered set with lt and E Q 5 then E is an ordered set with lt Difeifi nit io n Suppose 5 is an ordered set and E Q 5 o If there exists 3 E 5 such that X g B for all X E E then we say E is bounded above and call 3 an upper bound 0 If there exists 3 E 5 such that X 2 B for all X E E then we say E is bounded below and call 3 an lower bound Suppose Sis an ordered set and E Q S 0 Suppose there exists an 04 E S such that i a is an upper bound of E ii Ify lt a then 7 is not an upper bound of E We then say 04 is the least upper bound of E or the supremum of E and write 04 sup E 0 Suppose there eXists an 04 E S such that i a is a lower bound of E ii Ify gt a then 7 is not a lower bound of E We then say 04 is the greatest lower bound of E or the infimum of E and write 04 inf E Examples Lets consider the set Q with the standard ordering a LetXq QqZOandq l Examples Lets consider the set Q with the standard ordering a LetXq QqZOandq l X has a greatest lower bound and a least upper bound in X Examples Lets consider the set Q with the standard ordering a LetXq QqZOandq l X has a greatest lower bound and a least upper bound in X o LetXq Qqgt0and qlt1 Examples Lets consider the set Q with the standard ordering a LetXq QqZOandq l X has a greatest lower bound and a least upper bound in X o LetXq Qqgt0and qlt1 X has a greatest lower bound and a least upper bound in Q but not in X Examples Lets consider the set Q with the standard ordering a LetXq QqZOandq l X has a greatest lower bound and a least upper bound in X o LetXq Qqgt0and qlt1 X has a greatest lower bound and a least upper bound in Q but not in X o LetXq Qq20 Examples Lets consider the set Q with the standard ordering a LetXq QqZOandq l X has a greatest lower bound and a least upper bound in X o LetXq Qqgt0and qlt1 X has a greatest lower bound and a least upper bound in Q but not in X o LetXq Qq20 X has a greatest lower bound in X but is not bounded above Examples Lets consider the set Q with the standard ordering a LetXq QqZOandq l X has a greatest lower bound and a least upper bound in X o LetXq Qqgt0and qlt1 X has a greatest lower bound and a least upper bound in Q but not in X o LetXq Qq20 X has a greatest lower bound in X but is not bounded above 9 LetXnn Z Examples Lets consider the set Q with the standard ordering a LetXq QqZOandq l X has a greatest lower bound and a least upper bound in X o LetXq Qqgt0and qlt1 X has a greatest lower bound and a least upper bound in Q but not in X o LetXq Qq20 X has a greatest lower bound in X but is not bounded above 9 LetXnn Z X is not bounded above or below Examples Lets consider the set Q with the standard ordering a LetXq QqZOandq l X has a greatest lower bound and a least upper bound in X o LetXq Qqgt0and qlt1 X has a greatest lower bound and a least upper bound in Q but not in X o LetXq Qq20 X has a greatest lower bound in X but is not bounded above 9 LetXnn Z X is not bounded above or below 0 LetXq Q2 q2and q2 3 Examples Lets consider the set Q with the standard ordering a LetXq QqZOandq l X has a greatest lower bound and a least upper bound in X o LetXq Qqgt0and qlt1 X has a greatest lower bound and a least upper bound in Q but not in X o LetXq Qq20 X has a greatest lower bound in X but is not bounded above 9 LetXnn Z X is not bounded above or below 0 LetXq Q2 q2and q2 3 X bounded above and below but does not have a least upper bound or a greatest lower bound in Q Least Upper Bound Property Definition An ordered set 5 has the Least Upper Bound Property if for all E Q 5 such that o E is non empty o E is bounded above we have sup E exists in 5 quotr Bound Property e m39 Suppose 5 is an ordered set With the least upper bound property B Q 5 With B non empty and bounded below Let L be the set of all lower bounds of B Then asupL exists in 5 anda inf B In particular infB exists in 5 A Field IS a set F with two operations called addition denoted by and multiplication denoted by which satisfy the following field axioms A Axioms for addition A1 fXy FthenXy6 F A2 Addition is commutative For all Xy E F X y y X A3 Addition is associative For all Xyz E F xyz xyz A4 F contains a constant 0 such that for all X E F OX X A5 For every element X E F there is an element 7X E F such that X 7X O M Axioms for multiplication M1 fXy FthenXye F M2 Multiplication is commutative For all Xy E F X y y X M3 Multiplication is associative For all Xyz E F X y z X y 2 M4 F contains a constant 1 y 0 such that for all X E F 1 X X M5 IfX E F and X y 0 then there is an element 1X E F such that X 1X 1 M Axioms for multiplication M1 fXy FthenXye F M2 Multiplication is commutative For all Xy E F X y y X M3 Multiplication is associative For all Xyz E F X y z X y 2 M4 F contains a constant 1 y 0 such that for all X E F 1 X X M5 IfX E F and X y 0 then there is an element 1X E F such that X 1X 1 D The distributive law Xyzxyxz for all Xyz E F Examples of Fields The following are some examples of fields 0 The rational numbers Q o The real numbers R o The Complex numbers C o Integers mod a prime p Zp Examples of Fields The following are some examples of fields 0 The rational numbers Q o The real numbers R o The Complex numbers C o Integers mod a prime p Zp The following are not fields 0 The non negative rational numbers X E Q X 2 O o Integers mod a composite n Zn Theorems The axioms of addition imply the following a Ifxyxz thenyz b Ifxyx theny0 c Ifx y 0 then X 7y d If77x X Theorems The field axioms imply the following for all Xy7 z E F a 0X O b lfxy O andyy O thenxyy O C If 7Xy Xy Xy d XX Xy Ordered Field Definition An Ordered Field is a field F which is also an ordered set with lt such that o xyltxzifX7y726Fandyltz oxygt0ifxy FxgtOandygt0 If X gt 0 then we say X is positive Examples of Ordered Field The following are some examples of ordered fields a The rational numbers Q a The real numbers R Examples of Ordered Field The following are some examples of ordered fields a The rational numbers Q a The real numbers R The following are fields which can not be made into ordered fields 0 The Complex numbers C o Integers mod a prime p Zp Theorems The following are true in every ordered field a If X gt 0 then 7X lt O and vice versa Ifxgt0andyltzthenxyltxz c fxlt0andyltzthenxygtxz Ifx 7 0 then X2 gt 0 In particular 1 gt O Lemma 1 Let a lirn supzn Suppose a ioo Then there exits a subse quehce 0f such that xi 7 a for n 7gt 00 That is a is a cluster point of In Proof Let C is a cluster piont of By de nition a SupCi By Proposition 1327 V6 gt 07 there 30 E C such that c gt a7 6 Now let en i we inductively construct subsequence Ii such that for all n7 1 l a77ltzinlta7i n n For n 17 By Proposition 1327 there exits a cl 6 C such that a 7 12 lt cl S a Since cl is a cluster point of In there exists an Iii such that lzil 701 lt 1 2 Then we have lzil 7 al lt 1 why Now suppose that we have constructed Iii l l l Ii such that 1 i1lti2mltin 2i 7 al lt Let us construct In such that in lt in1 and lzin 7 al lt 1 ln fact7 by TH Proposition 132 again7 there exists a cn1 E C such that l c 7 7 lt c lt a 2n 1 n1 Since cn1 is a cluster point of In7 we see there are in nitely many In such that In 7 cn1l lt T en we can select an in1 such that in lt in1 and lzin 7 cn1l lt Therefore 1 lzin1 al S lzml Cn1l lt lCn1 Cl lt x By induction7 we then can construct a subsequence such that 75 lt Thus converges to a D Discontinuilies gt nn Mcan Definition Definition If f X a Y and f is not continuous at X E X then we say f is discontinuous at X Definition Definition Let f be defined by 37 b Consider any point X such that a g X lt b We write fX q if ftn a q as n a 00 for all sequences tn 6 X7 b such that thX Similarly we define f0 q if ftn a q as n a 00 for all sequences tn 6 37X such that thX Discontinuity Theorem ff is any function With domain 37 b then limtox ft exists if and only iffx fxi lithX ft Definition Let f be a function defined on 37 b If f is discontinuous at X and if fx and fxi exist then f is said to have a discontinuity of the first kind or a simple discontinuity at X Otherwise the discontinuity is said to be of the second kind There are two ways a function can have a simple discontinuity either fx 7 fxi in which case the value of fx is immaterial or fx fxi 7 fx Example a Define fX 1 if X is rational and O ifX is irrational Then f has a discontinuity of the second kind at every point X since fX nor fX Define fX X ifX is rational and 0 if X is irrational A o v Then f is continuous at X O and has a discontinuity of the second kind at every other point Define fX sin1X if X 7 O and O ifX 0 A r v Since neither f0 nor f07 exists f has a discontinuity of the second kind at X 0 mini 39 Monolonic Functions Monotonically Functions Definition Let f be real on 37 b Then f is said to be monotonically increasing on 37 b if a lt X lt b implies fx fy If the last inequality is reversed we obtain the definition of a monotonically decreasing function The class of monotonic functions consists of both the increasing and decreasing functions L Monolonlc Funcllons LeftRight Limits and Monotonic Functions Theorem Let f be monotonically increasing on a7 b Then fx and fxi exist at every point ofx ofa7 b More precisely 5 Palttltxft xii g 1 00 g fX infxlttltbft Furthermore ifa lt X lt y lt b then fx S W7 Corollary Monotonic functions have no discontinuities of second kind mini 39 Monolonic Functions Countany Many Discontinuities Theorem Let f be monotonic on 37 b Then the set of points of 37 b at Which f is discontinuous is at most countable Notice that the discontinuities of a monotonic function need not be isolated In fact given a countable subset E of 37 b which may even be dense we can construct a function f monotonic on 37 b discontinuous at every point of E and at no other point of 37 b Mini 39 Monolonic Functions Countany Many Discontinuities To see this let the points of E be arranged in a sequence Xn and let en be a sequence of positive numbers such that Z cn converges For a lt X lt b define fx 2 C xnltx Where if there are no points of Xn to the left of X the sum is empty and equal to 0 Since the sum converges absolutely the order is immaterial L Monolonlc Functions Countany Many Discontinuities We can then check that a f is monotonically increasing on 37 b b f is discontinuous at every point of E and fxn 7 fxn7 cn c f is continuous at every other point of 37 b d For all a lt X lt b fxi fx We call a function satisfying d continuous from the left And we also have an analogous notion of continuous from the right Note on Uniform Convergence In this note we extend of Proposition 514 in a general seeting without assuming the continuity of Theorem 1 Let fnz be a sequence offunctz39ons de ned over A C R and 0 6 A Suppose fnx converges to fx uniformly and for all n can limmnmO fnx em39st Then limmnmO f and limnnor00 an eccz39st and lim fx lim an 14mg TLHOO Or another way to say this Theorem is If fnx converges to f uniformly then lim lim fnz lim lim fnx 14mg TLAPHX WHOO 14mg That is under the assumption of uniform convergence we can interchange the order of the limit Proof We rst prove that an is convergent by Cauchy criterion For any 6 gt 0 since fnx uniformly converges there exists an N gt 0 such that for all n m gt N and for all z E A lmoemmwg o For any l gt 0 note that limmnmO fl 11 Thus there exists an 61 gt 0 such that for any x 6 0 7 610 61 mm fall lt 2 Therefore for any xed n m gt N for any x 6 0 7 in ux 6mm where 6m min6m6n combining 1 and 2 we have 6 e e lam anl S lfmiamllfm fnllfnianl lt g g 639 3 Thus an is a cauchy sequence and converge to say a To see limmnmO fx a rst note that for any n gt N xed by 1 we have mlmmemmwg maoo Since is a continuous function7 we have we 7 MM 7 mm 7 mng new 7 mng was 7 fm96 lt4 6 3 For the same reason7 taking limit m 7 00 for 3 we have an i at S 6 5 Now for any x 6 0 7 6mm 5n7 combining 47 2 and 57 we have 6 E 46 We 7 a S Wye mm Wye 7 ani an 7 a lt g g e 3 That is lirn f a 17mg D Bounded Maps Definition A mapping fof a set E into RR is said to bounded if there is a real number M such that ifxi M for all X E E Continuity and Compactquot Cm y m can Continuous Maps of Compact Spaces Theorem Suppose f is a continuous mapping ofa compact metric space X into a metric space Y Then fX is compact Closed and Bounded Sets and Compact Spaces Theorem Iff is a continuous mapping ofa compact metric space X into RR then fX is closed and bounded Thus f is bounded Real Functions on Compact Metric Spaces Theorem Suppose f is a continuous real valued function on a compact metric space X and M suPpEX HP 77 preX HP Then there exists points p7 q E X such that fp M and fq In Here suppex fp is the least upper bound of fp p E X and infpex fp is the greatest lower bounded of fp p E X Continuous 1 1 mappings Theorem Suppose f is a continuous 1 7 1 mapping ofa compact metric space X onto a metric space Y Then the inverse mapping f 1 defined on Y by f 1fx x X E X is a continuous mapping of Y onto X Uniform Continuity Definition Let f be a mapping of a metric space X into a metric space Y We say that f is uniformly continuous on X if for every 6 gt 0 there exists 6 gt 0 such that dyfP7 fq lt 6 for all p and q in X for which dXp7 q lt 6 Continuity and Compactquot Cm y m can Uniform Continuity and Compact Metric Spaces Theorem Let f be a continuous mapping ofa compact space X into a metric space Y Then f is uniformly continuous on X Continuity and Compactquot Cm y 2nd to Non Compact Sets of Reals Theorem Let E be a noncompact set in R1 Then a There exists a continuous function on E Which is not bounded b There exists a continuous bounded function on E Which has no maXImum If in addition E is bounded then c There exists a continuous function on E Which is not uniformly continuous c d r Continuity and Connectedquot Continuity and Connected ness Theorem If f is a continuous mapping ofa metric space X into a metric space Y and ifE is a connected subset ofX then fE is a connected subset of Y Continuity and Connected ness Theorem Let f be a continuous real valued function on the interval a7 b If fa lt fb and ifc is a number such that fa lt c lt fb then there exists a point X 6 a7 b such that fx c Note this theorem does not have a converse e it is not always the case that fact that if For all c with fa lt c lt fb then there exists a point X 6 a7 b such that fx c that f is then continuous Di lrmizadcn 0r Taylor s Theorem Theorem 515 Suppose f is a real function on 3 b n is a positive integer f 1 is continuous on 3 b f t exists for every t 6 37 b Let 04 B be distinct points ofa7 b and define n71 k a Pt 2 f k tiozk k0 39 Then there exists a point X between 04 and B such that Wm B n n 3 PB 70 Di i39l39vrr Taylor s Theorem For n 1 this isjust the mean value theorem In general the theorem shows that f can be approximated by a polynomial of degree n 7 1 and allows us to estimate the error if we know bounds on lf xl Differentiation of VectorrValued Functions Differentiation of Complex Functions Definition 516 Our definition of differentiation applies to complex valued functions defined on 37 b If at m ifzm where f1 and f2 are real valued then we have at far mt Also f is differentiable at X if and only if both f1 and f2 are Differentiation of VectorrValued Functions Differentiation of Vector Valued Functions Definition 516 Let f be defined on 37 b taking values in R For any X 6 37 b f X is the point if there is one for which lim Mif X 0 tax fix If f f17 7 f with each f1 a real valued function then f1001 1X7fn X and f is differentiable at X if and only if f is differentiable at X for all 1 g i g n Differentiation of VectorrValued Functions Differentiation and Operations Theorem Suppose f and g are defined on 37 b are vector valued and differentiable at a point X 6 37 b Then H g f g are differentiable at X and a H g X f X 00 b fg X f Xgx 00 3 00 Taylor39s Themmm Differentiation ofVeclorrValued Functions Example For real X define fx eiX cosx l isinx This can be taken as the definition of complex exponentiation Then f x ieiquot 75inx l icosx Then we have f27r 7 f0 171 0 but lf xl 1 for all real X Hence the mean value theorem fails in this case Taylor39s Theworn Differentiation ofVeclorrValued Functions Example On the segment 01 define fX X and gX X l XzeiXQ Since leitl 1 for all real 1 we see that lim 7 1 XH0gX Next we have gX12X7 telX2 0 lt X lt1 X sothat 2X77 71gt771 2i 2 X X lg Xl Z Th rm Differentiation of VectorrValued Functions Example Hence f x 1 lt X g X lg X 2 X and so P Iim 700 0 an g X Hence in particular L39Hospital39s Rule fails in this case Differentiation of VectorrValued Funcllons Mean Value Theorem for Vector Value Functions Theorem 519 Suppose f is a continuous mapping of 37 b into R and f is differentiable in 37 b such that W7 7 f3N S b7 3WXi Convergent Sequences is r Definition A sequence pn in a metric space X7 d is said to converge if there is a point p E X with the following property Vs gt 03NVn gt Ndpr7 p lt e In this case we also say that pn converges to p or that p is the limit of pn and we write pn a p or limrH00 pn p If pn does not converge we say it diverges If there is any ambiguity we say pn convergesdiverges in X The set of all pn is said to be the range of pn which may be infinite or finite We say pn is bounded if the range is bounded Convergent Sequences is 7 ex uen Ce Cam Sequonca Exigm Notice that our definition of convergent depends not only on pn but also on X For example 1n n E N converges in R1 and diverges in 0 oo consider the following sequence of complex number ie X R2 If 5 1n then limrH00 5n 0 the range is infinite and the sequence is bounded If 5 n2 then the sequence 5 is divergent the range is infinite and the sequence is unbounded If 5 1 l 71 n then the sequence 5 converges to 1 is bounded and has infinite range If 5 i the sequence 5 is divergent is bounded and has finite range If 5 1 n 12 3 then 5 converges to 1is bounded Properties of Convergent Sequences Theorem a pn converges to p E X if and only if every neighborhood of p contains pn for all but finitely many n b lfp7p E X andifpn converges to p and to p then p p c lfpn converges then pn is bounded d le Q X and ifp is a limit point of E then there is a sequence pn in E such that p IimrH00 pn em Sequences Properties of Convergent Sequences Theorem Suppose 5n7 tn are complex sequence With IimrH00 5n s and IimrH00 tn t Then a Iimnaoosn tn s t b IimrH00 6 5 6 5 and IimrH00 c 5n c s for any number 6 c IimrH00 sntn d Iimnaooi st me H Properties of Convergent Sequences Theorem a Suppose xn E Rkn E N and xn 041 7 704m Then xn converges to x 0417 Wok ifand only if Iim 09 Ozj 1 ltjlt k naoo b Suppose xn7 yn are sequences in RR n is a sequence of real numbers and X a x7yn a y n a 3 Then Iim xnyn xy naoo n39ingxwyn Xy Iim nxn Bx naoo a 5 eq Hen ces Subsequences Definition Given a sequence pn consider a sequence nk of positive integers such that n1 lt n2 lt n3 lt Then the sequence pm is called a subsequence of pn If pm converges its limit is called a subsequenta limit of pn It is clear that pn converges to p if and only if every subsequence of pn converges to p a 5 eq Hen ces Subsequences and Compact Metric Spaces Theorem a Ifpn is a sequence in a compact metric space X then some subsequence ofpn converges to a point ofX b Every bounded sequence in RR contains a convergent subsequence L setman s equences Subsequences Limits Theorem The subsequential limits ofa sequence pn in a metric space X form a closed subset of X Cauchy Sequence Definition A sequence pn in a metric space X7 d is said to be a Cauchy sequence if for every 6 gt 0 there is an integer N such that dpnpm lt e for all I77 I77 2 N Definition Let E be a nonempty subset of a metric space X7 d and let 5 dp7 q p7 q E E The diameter ofE is sup 5 If pn is a sequence in X and if EN consists of the points pN7pN17 it is clear that pn is a Cauchy sequence if and only if im diamEN O Naoo Cauchy Sequences and Closed Sets Theorem a If is the closure ofa set E in a metric space X then diam F diam E b If Kn is a sequence of compact sets in X such that Kn C Kn1 n E N and iflimrH00 diam Kn 0 then Kn consists of exactly one point Cauchy Sequences Cauchy Sequences and Convergent Sequences Theorem a In any metric space X every convergent sequence is a Cauchy sequence b IfX is a compact metric space and if pn is a Cauchy sequence in X then pn converges to some point ofX c In RR every Cauchy sequence converges Cauchy Sequences Complete Spaces Definition A metric space is said to be complete if every Cauchy sequence converges Notice that all compact metric spaces are complete but there are metric spaces like RR which are complete but not compact Lemma Every closed subset ofa complete metric space is complete Cauchy Sequences IncreasingDecreasing Sequences Definition A sequence 5 of real numbers is said to be a monotonically increasing if sn lt sn1 for all n E N b monotonically decreasing if sn 2 sn1 for all n E N c monotonic if it is monotonically increasing or monotonically decreasing Theorem Suppose 5 is monotonic Then 5 converges if and only if 5 is bounded Limit Points of Sequences of Functions Theorem 711 Suppose a f uniformly on a set E in a metric space Let X be a limit point of E and suppose that tlmfn An n 123 Then An converges and W Hugo An Theorem 712 If fn is a sequence of continuous functions on E and if fn a f uniformly on E then f is continuous on E Unironn Convergence and Continuity nw In I39In Collv Limit Points of Sequences of Functions Theorem 713 Suppose K is compact and a fn is a sequence of continuous functions on K b fn converges pointWise to a continuous function f on K c fnx 2 fn1x for allx E K n 172737 Then fn a f uniformly on K Unironn Convergence and Continuity mu In I39In Collv Space of Continuous Functions Definition 714 If X is a metric space CX will denote the set of all complex valued continuous bounded functions with domain X Note that boundedness is redundant ifX is compact Space of Continuous Functions Definition 714 We associate with each f E CX its supremum norm llfll SUPxexlflxl Since f is assumed to be bounded lt 00 It is obvious that 0 only if fx O for every X E X that is only iff0 lfhfg then lhXl S lfXl lgXl S llfll llgll for all X E X hence llfgll llfllllgll Space of Continuous Functions Definition 714 If we define the distance between fg E CX as llf igll then this makes CX into a metric space The previous theorem can be restated as saying Theorem A sequence fn converges to f With respect to the metric on CX ifand only if fn a f uniformly on X Accordingly closed subsets of CX are sometimes called uniformly closed and the closure of A Q CX is called its uniform closure Theorem 715 CX is a complete metric space Integration of Sequences of Functions Theorem 716 Let a be monotonically increasing on 3 b Suppose 6 7201 on 3 b for n 1 2 3 and suppose fn a f uniformly on 3 b Then f E 7201 on 3 b and b b fdaimfnda The existence of the limit is part of the conclusion Integration of Sequences of Functions Corollary If 6 7201 on 37 b and if fx anx a g X g b n1 With the series converging uniformly on 37 b then b 00 b fdaZ fnda a quot13 Differentiation of Sequences of Functions Theorem 717 Suppose fn is 3 sequence of functions e3ch differentiable on 3 b and such that fnxo converges for some point X0 on 3 b Iffr converges uniformly on 3 b then fn converges uniformly on 3 b to 3 function f and W quotingofn m a x b Nowhere Differentiable Function Theorem 718 There exists a real continuous function on the real line Which is nowhere differentiable Limits of loo Definition Let 5 be a sequence of real numbers If For all real M there is an integer N where 5 2 M whenever n 2 N then we write 5 gt 00 For all real M there is an integer N where 5 2 M whenever n g N then we write 5 gt foo Upper and Lower Limits Definition Let 5 be a sequence of real numbers Let E be the set ofx in the extended real number system such that 5W a X for some subsequence 5W So E has all subsequential limits of 5 plus possibly foo or 00 Let 5 sup E 5 inf E 5 is the upper bound of 5 and 5 is the lower bound of 5 We write lim sup 5 5 naoo lim inf 5n 5 naoo t 12 gum 5 Properties of Lim Sup and Lim Inf Theorem Let 5 be a sequence of real numbers let 5 IimrH00 sup 5 and let E be the set ofsubsequential limits of 5 Then a 5 E E b lfx gt 5 then there is an integer N such that n 2 N implies 5 lt x Further 5 is the only extended real number satisfying properties a and There is an analogous result for 5 IimrH00 infsn Examples a Let 5 be a sequence containing all rational numbers Then every real number is a subsequential limit and lim inf sn7oo7 lim sup snoo naoo naoo 1 n b Let 5 lm Then lim inf sn717 lim sup sn1 naoo naoo c For a real valued sequence 5 with limrHOO 5n 5 if and only if lim inf 5n lim sup 5 s naoo naoo Upper and Lower I Star Lim u Ordering of Lim SupLim Inf Theorem Ifsn tn for n 2 N Where N is fixed then Iim infsn lt Iim inftn naoo naoo Iim sup 5 Iim sup tn naoo naoo U m ow Some Special Sequences Squeeze Theorem We now compute the limits of some special sequences The proofs are based on the following lemma Lemma IfO Xn 5 for n 2 N Where N is some fixed number and 5 a 0 then tn a O as well Um Some Speclal Sequences Special Limits Theorem a pr gt 0 then IimrH00 quot17 7 O b pr gt 0 then IimrH00 36 1 c IimrH00 W 1 d pr gt O anda is real then IimrH00 12 0 e Ifixi lt 1 then IimrH00 X O A complex number is an ordered pair 37 b where a and b are real numbers Here an ordered pair means that 37 b 7 b7 3 if a 7 b Suppose X 37 b and y 7 d are complex numbers Then we sayxy ifand only ifacand bd Wedefine xyacbd Xyacibdadbc Complex Numbers Are A Field 7 The complex numbers With this addition and multiplication are a field For any real numbers 3 and b we have 30 b 0 3 b0 30 b 0 3b 0 In particular this means that if we identify 30 with 3 then the field of real numbers is a subfield of the complex numbers Definition Let i 01 271 Ifa and b are real numbers then 37 b a i b i 9 Complex Field Misc guns Complex Conjugate Definition If a and b are real numbers and z a bi then the complex number 2 a 7 bi is called the complex conjugate of z The numbers a and b are also called the real part and the imaginary part of 2 respectively We sometimes write aRez7 b mz Complex Conjugate Results The Complex Fiel M D ei fi n it iorn39 If 2 is a complex number its absolute value lzl is the non negative square root of z zz Note that the absolute value is well define because 22 is always non negative and we have shown that there is always a unique positive square root of any non negative number Th eo re m Ifxis real thenYx and lxi xp So lxl X ifXZ O and lxl 7X ifxlt 0 Absolute Value Results The Complex Fiel M Definition If X17X2 Xn are complex numbers we write n X1X2XnZXj j1 Theorem Schyvarz Inequality If 31 73 and b17 7 bn are complex numbers then n n n Zaj z E 2 W2 MW 11 11 11 Misc Results isc Theorem Let aoz alzrquot1 an 0 be a polynomial equation With real coefficients If the complex number 2 a bi is a root of this equation then so is its complex conjugate f a 7 bi

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