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# Class Note for MATH 1432 at UH 3

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This 7 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15
Lecture 17Section 101 Least Upper Bound Axiom Section 102 Sequences of Real Numbers Jiwen He 1 Real Numbers 11 Review Basic Properties of R R being Ordered Classi cation 0 N 017 2 i i natural numbers 0 Z 11177277170717271117 integers 39 Q 1 P74 E Z q 0 rational numbers 0 R real numbers Q U irrational numbers WAi R is An Ordered Field ozSyandySz zgzi ozSyandySz 42gt zyi VLyER zgyorygzi ozSyandzER zz yzi OIZOandyZO zyZOi Archimedean Property and Dedekind Cut Axiom Archimedean Property VI gt 07 Vy gt 07 3n 6 N such that nz gt y Dedekind Cut Axiom Let E and F be two nonempty subsets of R such that o EUFR 1 oE F onEEVy FWehavex y Then7 Hz 6 lR such that xgz VIEE and zgy VyEF Least Upper Bound Theorem Every nonempty subset S of lR With an upper bound has a least upper bound also called supremum 12 Least Upper Bound Basic Properties of lR Least Upper Bound Property De nition 1 Let S be a nonempty subset S of lR o S is bounded above if 3M 6 lR such that x g M for all x E S M is called an upper bound for S o S is bounded below if 3m 6 lR such that x 2 m for all x E S m is called an lower bound for S o S is bounded if it is bounded above and below Least Upper Bound Theorem Every nonempty subset S of lR With an upper bound has a least upper bound also called supremum Proof Let F upper bounds for S and E lR E E7 F is a Dedekind cut 3b 6 lR such that x g b7 V95 6 E and b g y7 Vy E F bis also an upper bound of S i b is the lub of S Supremum or In mum of a Set S De nition 2 Let S be a nonempty subset of lR With an upper bound We denote by supS or lubS the supremum or least upper bound of S M e Theorem 3 Let M supS Then 3L oxSM VxES 0V6gt07 MigMMWSy Q De nition 4 Let S be a nonempty subset of R with a lower bound We denote by infS or glbS the or greatest lower bound of S E j m m Theorem 5 Let m infS Then 0 2 2 m Va 6 S 0V6gt07 mme S 0 Examples Supremum or In mum of a Set S Examples 6 0 Every nite subset of R has both upper and lower bounds sup123 37 inf123 1 o If a lt b7 then b supab supab and a infab infab o IfSq Qeltqlt7rtheninfSesupS7r o IfS7z R7c2lt71397 theninfS7 supS o IfSz6Q2232lt77theninfS7 supS Theorem 7 The notions of and supremum are dual in the sense that infS isup7S where 7539 73 s 6 S 2 Sequences of Real Numbers Sequences De nition De nition 8 A sequence of Teal numbers is a real valued function de ned on the set of positive integers Nquot Nquot 12 9ngt gtanfn ER Where the nth term is denoted by an ZeX The sequence a17 a27 denoted by an 1 or an 7 1s Examples 9 0 an i n E Nquot is the sequence 17 5 7 n i 1 2 3 4 i 0 an 7 71 n EN 71s the sequence 57 71737 0 an n27 n E Nquot is the sequence 17 4 97 167 0 an cosn7r 71 7 n E Nquot is the sequence 71 17 71 Limit of a Sequence De nition 10 Let un il be a sequence of real numbers A real number L is a limit of un il denoted by L lim an ngtoo iblxamples 11 V6 1f WEN W Nh lln l39inilglg aVJBON For any 6 gt 0 given choose N gt 0 such that 6N gt 1 iiei lt s Then if n gt N we 1 1 have0ltmltmlts 0 If an 7amp1 n E Nquot then limnn00 an 1 For any 6 gt 0 given choose N gt 0 such that 6N gt 1 ie lt s Then if n gt N we have 0lt 7 1 1 1 7 1 7 E lt N lt so Convergent Sequence De nition 12 o A sequence that has a limit is said to be convergent o A sequence that has no limit is said to be divergent Uniqueness of Limit 1f limnn00 an L and limnn00 an M then L Mi Proof V6 gt 0 EN gt Osuch that laniLl lt 5 and laniMl lt Vngt Ni lLiMl S laniLllaniMl lt6 LMl Example 13 0 if an cosnrr 71 n E Nquot the sequence on is diver gent 1f 6 then the interval 1 7 z has a length g that is lt 1 VI 6 R it can not contain 1 and 71 at the same time Therefore it is not possible to nd N such that law 7 11 lt if n gt Ni Boundedness of a Sequence De nition 14 A sequence un il is bounded above or bounded below or bounded if the set S a1 a2 i i is bounded above or bounded below or bounded Examples 15 0 If an f n E Nquot then the sequence an is bounded above by M 2 1 and bounded below by m S 0 0 If an cosnrr 71 n E Nquot then M 2 1 is an upper bound for the sequence an and m S 71 is an lower bound for the sequence on Theorem 16 Every convergent sequence is bounded limnnman L V6 gt 0 EN 6 Nstl laniLl lt 6Vngt N lanl S laniLllLlltelLLVngtN donel Theorem 17 Every unbounded sequence is divergent Monotonic Sequence De nition 18 o A sequence a il is increasing if an S an1 for all n E N o A sequence a il is decreasing if an 2 an1 for all n E N Theorem 19 o A bounded increasing sequence conuerges to its lub o a bounded decreasing sequence conuerges to its glb Examples 20 o If an 71 n E N then an is decreasing bounded and linonu00 an infan 0 lt OLE 1 Z E lt 1 o If an n2 n E N then an is increasing but unbounded aboue therefore is diuergent Example H I I I I I I I I I I I I I I I I l o 2 a1 2 U3 as J J J IIJH nnml I I I I I I I 1 3 3 i 1 1 2 3 4 5 6 7 u 2 3 4 5 Erample 21 Let an n E N 39 39 39 Ctr11 n1 n1 i n22n1 0 an is increasing lt an i n2 n i n22n gt 1 1 2 3 99 i o The sequence displays as 5 3 Z W gt supan 1 and 1nfan 5 gt lim an supan 1 I LHOO Example Z a4 3 I I I I T 9 1 2 3 4 5 6 n Example 22 Let an with n nn7 11I an1 7 2 1 n 7 2 a an1s decreasmg c an 7 n D27 7 m lt 1 I supan 2 and infan 0 gt lim an infan 0 mace Example y i 1 a1 6X 6 2 a2 3 1 a3 4 a4 1 2 3 4 Example 23 Let an 0 an is decreasing4e Let lex ea 690 1 a 0 f x 600 lt o supan i and infan 0 gt hm an infan 0 Example Example 24 Let an 71 n 17 27 0 an is decreasing for n 2 3 lex Let ailac 61001n00 19X fm 61wgtlm1x1nx lt 0 o infan 1 gt hm an infan 1 7100 Outline Contents

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