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Analysis I

by: Zechariah Hilpert
Zechariah Hilpert
GPA 3.8


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This 2 page Class Notes was uploaded by Zechariah Hilpert on Thursday September 17, 2015. The Class Notes belongs to MATH 521 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/205290/math-521-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.

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Date Created: 09/17/15
Chapter 1 HOW TO DO PROOFS There is a certain formula or method to doing proofsi Some of the guidelines are given belowi The most important factor in learning to do proofs is practice7 just as when one is learning a new language 1 There are very few words needed in the structure of a proof Organized in rows by synonyms they are To show Assume Let Suppose Define If Since Because By Then Thus So There exists There is Recall We know But 2 The overall structure of a proof is a block structure like an outline For example To show If A then B and C Assume A To show a B b C a To show B Thus B b To show C Thus C So C So if A then B and C 3 Any proof or section of proof begins with one of the following a To show If A then B b To show There exists C c To show such that D Immediately following this7 the next step is Case a Assume the ifs and to show7 the thensi The next lines usually are To show B Case b To show an object exists you must nd it e next lines usually are efine xxx To show xxx sat isf ies D Case a Rewrite the statement in E e next line is usually To show E by using a de nition A useful guideline is7 Don t think too much77 Following the method usually produces a proof without thinking Most of doing proofs is simply rewriting what has come just before in a different form by plugging in a de nition There are some kinds of proofs which have a special structurel Proofs by contradiction Assume the opposite of what you want to showl End up showing the opposite of some assumption not necessarily the assumptionl Contradict ion Thus Assumption is wrong and what you want to show is true Counterexamples To show that a statement7 If then To show There exists a such that a it satis es the ifs of the statement that you are showing is false b it satis es the opposite of some assertion in the thens of the statement that you are showing is false 77 is false you must give an example Proofs of uniqueness To show that an object is unique you must show that if there are two of them then they are really the same To show A THING is unique Assume X1 and X2 are both THINGS To show X1 X2 Proofs by induction A statement to be proved by induction must have the form B for all positive integers n The proof by induction should have the form Proof by induction Base case To show If A then B for n 1 Thus if A then B for n 1 Induction step then B for nlt N To show If A then B This last to show line contains exactly the same statement except with n replaced by N and for all positive integers n removedl


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