Foundations of Analysis
Foundations of Analysis MATH 4200
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