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# Foundations of Analysis MATH 4200

Utah State University

GPA 3.77

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This 5 page Class Notes was uploaded by Darren Schulist on Wednesday October 28, 2015. The Class Notes belongs to MATH 4200 at Utah State University taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/230423/math-4200-utah-state-university in Mathematics (M) at Utah State University.

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Date Created: 10/28/15

Math 4200 Review 1 1 Show that x 43 is irrational 2 Write the following statements in the form p 9 q Find the converse and contrapositive for each a Whenever f x lt g x lt b A necessary condition for a function f to be differentiable at x a is for f to be continuous at x a 3 Construct atruth table for the statement p and N r 9 N q or r 4 A sequence an is said to be a Cauchy sequence iffor each 6 gt O there exists NE gt 0 such that for all positive integers m and n if n m 2 NE then l an 7 aml lt e State the precise negation of this statement 5 In a class with 30 students how many ways can you select 5 students to serve on a committee given that three of the students refuse to serve together Each one of these two students is willing to serve but not if one of the others is selected 6 Prove that n3 5n is divisible by 6 for each n E N 7 Give an example of a finite field Describe the addition and multiplication operations for your field 8 Let F be an ordered eld Prove each of the following properties ab0 implies a0 or b0 if OSa and OSb then OSab 8 Show that the following sets are equivalent that is for each pair of sets there exists a 11 correspondence between them a O1and744 b 01and 000 c QandJ 9 Let S be the set of all points in the upperhalf plane with integer coordinates That is S Xy X is an integer and y is a positive integer Show that S is equivalent to 10 Show that the set of all sequences of 3 s and 739s is not a countably in nite set Math 4200 Review 2 1 De nitions a List the axioms for the real line b State the Dedekind Principle c What is a sequence d Suppose 5 is a sequence De ne what it means for 5 to have limit 5 e Suppose 5 is a sequence De ne what it means for 5 to have limit 00 f Suppose 57 is a sequence De ne what it means for 57 to be a Cauchy sequence 0 g De ne what it means for the series Z a to converge 11 2n 72 3n71 339 2 Using the limit de nition prove that lim TLHOO 3 Let S be a non empty subset of 9 and suppose S is bounded above Let m supS Show that for each 6 gt O there exists x in S such that m 7 e lt x g m This is the quotbackaway principlequot for suprema 4 Suppose the sequence bu is bounded That is there exists M gt 0 such that bn M forall n If lim an O showthat lim anbn O 5 Suppose 5 is a monotone non increasing sequence that is for each n 5W1 5 Suppose also that 57 is bounded below that is there exists a real number M such that for all n M g 5 Prove that 57 converges 6 State and prove the Squeeze Theorem for sequences 7 Suppose the sequence an converges to L Show that an is bounded that is 3 MgtOsuchthatlanl ltM VnEJ 00 De nition De ne what it means for a function f to have a limit L at 1 a The notation is lim L 3H 50 State the Nested Intervals Theorem 10 State the BolzanoWeierstrass Theorem 11 Suppose the sequence an converges to L Show that an is a Cauchy sequence 12 Suppose the sequence an is a Cauchy sequence Show that an converges 13 Use the limit de nition to show that lin 1 2 14 Suppose f is de ned on 00 00 There are essentially three different ways for f to not have a limit at 1 a State the three different ways and give a speci c example of each 15 De ne what is meant by lifn 16 Suppose lim L lim h1 L7 and for each X 3 91 3 Show that lim 91 L 17 Suppose lim O and 91 is bounded 3 M gt 0 such that l91lltM V169 Showthat lim 0 18 State and prove one of the limit theorems for functions 19 De ne what it means for a function f to be continuous at 1 a What does it mean for f to be continuous on an interval 17 b 20 Properties of continuous functions a X o b locally bounded c 11 gt monotone d Fixed Point Theorem e Max Min Theorem f Intermediate Value Theorem g characterization of continuity in terms of convergent sequences

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