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# Foundations of Real Analysis MATH 6101

UNCC

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This 1 page Study Guide was uploaded by Mrs. Dangelo Fahey on Sunday October 25, 2015. The Study Guide belongs to MATH 6101 at University of North Carolina - Charlotte taught by Staff in Fall. Since its upload, it has received 35 views. For similar materials see /class/228928/math-6101-university-of-north-carolina-charlotte in Mathematics (M) at University of North Carolina - Charlotte.

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Date Created: 10/25/15

MATH 6101090 FOUNDATIONS OF REAL ANALYSIS Fall 2004 Study Guide for the Final 1 De nitions and axioms to remember H From the yellow book De nition 33 absolute value 41 minimum and maximum 42 upper and lower bounds and 43 supremum and in mum Axiom 44 completeness De nition 71 convergence 98 00 as a limit 101 monotone sequences 106 lim inf and limsup 108 Cauchy sequences 111 subsequences 116 subsequential limits 141 summation notation 142 in nite series 143 Cauchy criterion for series E0 From your notes accumulation point of a set 2 Theorems you should remember 1 With proof Theorem 35iii triangle inequality remember also the generalization in Exercise 36 Corollary 45 existence of inf Properties 46 and 47 Archimedean property and denseness of Q us ing the Completeness Axiom Theorem 91 convergent sequences are bounded Theorems 9 93 and 96 what happens to the limit when you multiply by a constant add or divide two sequences Exam ples 97ab Theorem 10 monotone bounded sequences converge 104 monotone unbounded goes to ioo Corollary 105 Lemma 109 convergent sequences are Cauchy sequences Lemma 1010 Cauchy sequences are bounded Theorem 113 every sequence has a monotonic subsequence Theorem 144 Cauchy criterion for series only equivalence with sequential Cauchy criterion Example 141 geometric series Corollary 145 terms in a convergent series go to zero 148 Ratio Test proof only of weaker form in Exercise 912a to Without proof Theorems 94 95 99 and 910 limit of product and inverse of a sequence Examples 97cd Theorem 10 limit exists iff limsup is same as liminf Theorem 1011 Cauchy sequences converge Theorem 112 subsequences have the same limit Corollary 114 equivalent de nition of limsup and liminf Theorem 115 Bolzano Weierstrass Theorem 11 set of subsequential limits Theorem 121 lim sup of a product Example 142 2211np 146 Comparison Test 147 Corollary absolute convergent implies convergent 149 Root test 3 What to expect The exam will be closed book you may use the same handouts as for the midterm The above guide is meant to help with the mandatory part For the optional part prepare as if it was another midterm The mandatory part will be as long as the midterm the optional part will have only about 5 questions Besides remembering the de nitions theorems and proofs above you Should be prepared to calculate the limit of a Simple sequence or series using the de nition of convergence only or the limit of a more complicated sequence or series using all the results we learned 1 could also ask you to determine the supremum the in mum and the accumulation points of a set or the set of subsequential limits of a sequence Study Guide for the Final

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