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Unformatted text preview: MATH 104, SUMMER 2008, REVIEW SHEET FOR FINAL EXAM The final will be on Thursday, August 13, 2:10-4:00pm . It is cumulative (covering the whose semesters materials) but with an emphasis on 2034 . Approximately 50% of the exam problems are based on these sections discussed after the second midterm. To do well on this exam you should be able to do at least each of the following 1 : Section 1: The set N of natural numbers (1) State the principle of mathematical induction. (2) Use mathematical induction to prove an elementary fact about natural num- bers. (For example, a sequence given by some inductive formula is mono- tonic). Section 2: The set Q of rational numbers (1) Define algebraic numbers. (2) State the rational zeros theorem. (3) Use the rational zeros theorem to show that specific algebraic numbers are not rational. (For example, p 1 + 3 5 ) Section 3: The set R of real numbers (1) List the Field Axioms. (2) List the ordering axioms. (3) Use the ordered field axioms to prove some simple facts about ordered fields. (For example, the additive cancelation law, or ( a F )(0 < a 2 ) ) (4) Define absolute value. (5) State (and use in proof) any version of the triangle inequality. Section 4: The Completeness Axiom (1) Define max, min, upper bound, lower bound, sup, and inf. (2) Give examples of sets having various combinations of the above terms. (For example, a set with a sup and a min, but no max, etc. . . ). (3) State the completeness axiom. (4) State (and use in proof) the Archimedean property. (5) State and use in proof the denseness property of Q . Section 5: The symbols + and- (1) Use the symbols appropriately. Section 7: Limits of Sequences (1) Define lim n s n = L . (2) Define converge and diverge. Section 8: A Discussion about Proofs (1) Give a precise proof that a given sequence converges to a given limit. (For example, show ( 3 n +2 2 n +3 ) n N converges to 3 2 ). (2) Give a precise proof that a given sequence does not converge. (For example, show 3 n 2 +2 2 n +3 n N diverges). Section 9: Limit Theorems for Sequences (1) Prove that every convergent sequence is bounded. 1 This review guide was summarized by Benjamin Johnson, a Berkeley math GSI. 1 2 MATH 104, SUMMER 2008, REVIEW SHEET FOR FINAL EXAM (2) Prove one of the limit laws. (For example, if ( s n ) s and ( t n ) t , then ( s n + t n ) s + t ). (3) Use the limit laws to compute the limit of a given sequence converges, and to justify that the sequence converges to that limit. For example, s n = n- 2 6 n 2 +5 ). (4) Define lim n s n = or lim n s n =- . (5) Give a precise proof that a given sequence diverges to or- . (For exam- ple, ( n 2- 4) n N )....
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