} is said to be a Cauchy sequence provided the following condition is true.

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1 Math 4200, Fial Exam Review I. Itroductio to Proofs 1. Prove the Pythagorea theorem. 2. Show that 43 is a irratioal umber. II. Itroductio to Logic 1. Costruct a truth table for the statemet ( p ad ~ r ) ( ~ q or r ). 2. Write the followig statemets i the form p q. Fid the cotrapositive for each. a) Whe f '( x) 0 for each x i a iterval I, f is icreasig o I. b) A ecessary coditio for a fuctio to be differetiable at x a is for f to be cotiuous at x a. 3. A sequece { a } is said to be a Cauchy sequece provided the followig coditio is true. For each 0, there exists N 0 such that for all positive itegers m ad, if m, Nthe a am. State the precise egatio of this coditio. III. Coutig Argumets 1. State the Fudametal Coutig Priciple. 2. State the Pigeo-Hole Priciple. 3. I a class with 30 studets, how may ways ca you select 5 studets to serve o a committee give that three of the studets refuse to serve together? (Each oe of these three studets is willig to serve but ot if oe of the others is selected.)

2 IV. Itroductio to Sets 1. Defie each of the set operatios. 2. a) Suppose A, B, ad C are sets. Show that A ( B C) ( A B) ( A C). b) Let R deote the positive real umbers. Fid (5,13 ) ad (, ). R R 3. Show that the followig sets are equivalet; that is, for each pair of sets, there exists a 1-1 correspodece betwee them. a) (0,1) ad ( 4,4) b) (0,1) ad ( 4, ) c) Q ad J 4. Let S be the set of all poits i the upper-half plae with iteger coordiates. That is, S = { (x,y) : x is a iteger ad y is a positive iteger}. Show that S is equivalet to J. 5. Show that the set of all sequeces of 3's ad 7's is ot a coutably ifiite set. Describe how this result will imply that the set of all real umbers is a ucoutable set. V. Mathematical Iductio 1. List the axioms for the atural umbers. List the equivalet versios of the iductio axiom. 2. Prove the Fudametal Theorem of Arithmetic 3. Prove: For each atural umber, is divisible by Prove the Biomial Theorem VI. Fields 1. List the axioms for a field.

3 2. List the axioms for a ordered field. 3. Give examples of fiite fields ad ordered fields. VII. The Real Lie 1. Let A be a subset of the real lie R. Defie what is meat by the least upper boud of A, the greatest lower boud of A, the sup A, ad the if A. 2. State the completeess axiom for the real lie. 3. Show that the Least Upper Boud Priciple is equivalet to the Greatest Lower Boud Priciple. 4. State the Dedekid Priciple. 5. Show that the Dedekid Priciple is equivalet to the Least Upper Boud Priciple. 6. State ad prove the two backaway priciples. 7. Prove the Archimedea Property. VIII. Sequeces 1. What is a sequece? 2. Defie what it meas for a sequece to coverge. Notatio: lim a L or a L. 3. Prove that lim a L if ad oly if every ope iterval ( x, y ) cotaiig L must also cotai all but fiitely may terms of { a }. 4. Prove that 2 1 lim Show that every bouded, mootoe sequece must coverge. 6. Completely defie the real umber e. 7. Show that every coverget sequece is bouded. 8. State ad prove the basic limit theorems for sequeces.

4 9. State ad prove the Nested Itervals Theorem. 10. State ad prove the Squeeze Theorem for Sequeces. 11. State ad prove the Bolzao-Weierstrass Theorem. 12. Defie what it meas for a sequece { a } to be a Cauchy Sequece. 13. Prove that every Cauchy sequece is bouded. 14. Prove that every coverget sequece is a Cauchy sequece. 15. Prove that every Cauchy sequece must coverge. IX. Limits of Fuctios 1. Defie what is meat by a fuctio with domai D R ad rage R. 2. Defie precisely what is meat by sayig that the limit of f ( x ) as x approaches a is equal to L. The otatio is lim f ( x) L. 3. Defie precisely what is meat by lim f( x) ( ). x () 4. Use the limit defiitio to show that lim x 2. x 4 5. State ad prove the Squeeze Theorem for limits of fuctios. 6. State the limit theorems for fuctios. Prove two of them just for fu. X. Cotiuous Fuctios 1. Defie what it meas for a fuctio f to be cotiuous at x a. What does it mea for f to be cotiuous o a iterval I? 2. Show that a fuctio f is cotiuous at x a if ad oly if f preserves coverget sequeces at x a. 3. State the basic theorems for cotiuous fuctios. Prove all 17 of them just for fu.

5 4. Prove: If f is a cotiuous fuctio o [ ab, ] the f assumes a maximum value ad a miimum value o [ ab, ]. 5. Prove that the fuctios defied by f ( x) si x ad g( x) cos x are cotiuous. 6. Suppose lim f( x) 0 ad the fuctio g is bouded. Prove that lim f( x) g( x) Prove the Itermediate Value Theorem. XI. Differetiable Fuctios 1. What does it mea for a fuctio f to be differetiable at x a? What does the derivative of f at x a represet? 2. Suppose f is differetiable at x a. Prove that f is cotiuous at x a. 3. From a graphical poit of view, what is the differece betwee a cotiuous fuctio ad a differetiable fuctio? 4. State ad prove the Liear Approximatio Theorem. Why is it importat? 5. State ad prove each of the basic theorems of differetial calculus. 6. State ad prove the Mea Value Theorem. Why is it importat? 7. Suppose f '( x) 0 for each x i the iterval (,) rs. Prove that f is strictly icreasig o (,) rs. 8. State both defiitios for the defiite itegral. 9. State ad prove the Fudametal Theorem of Calculus. XII. Why should you study Calculus?

MAS111 Convergence and Continuity

MAS111 Convergence and Continuity MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece