370_Notes_& solution

370_Notes_& solution - Notes for Math 370, Spring 2008...

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Unformatted text preview: Notes for Math 370, Spring 2008 Eric Hayashi San Franciso State Univerity May 5, 2008 Abstract 1. Real Number System R 1. properties of the real numbers 1. de&ciencies with Q 2. ordered &eld properties 3. upper and lower bounds, inf S , sup S 4. Axiom of Completeness 5. Archimedian property 6. density of Q 2. sequences of real numbers 1. subsequence 2. limit of a sequence 3. Limit Laws (properties of limits) 4. Monotone Limit Theorem 5. Bolzano-Weierstrass Theorem 6. lim inf ; lim sup 7. Cauchy sequence, Cauchy completeness 2. Real-valued functions 1 1. limits 2. sequential characterization of limits 3. continuity 4. uniform continuity 5. Extreme Value Theorem 6. Intemediate Value Theorem 7. Inverses of continuous functions 3. Di/erentiation 1. the derivative 2. Mean Value Theorem and its consequences 3. L&Hopital&s Rule 4. The Riemann integral 1. Partitions, upper and lower sums, upper and lower integrals, Riemann sums 2. integrability 3. properties of the integral 4. The Fundamental Theorem of Calculus 5. rigorous development of logarithmic, exponential, and trig functions 5. Additional topics as time permits 1. topology of the real line 2. Heine-Borel theorem 3. metric spaces 1. The Real Number System The 19th century mathematician Leopold Kronecker said, " God created the inte- gers, all the rest is the work of mankind." tor all x;y and z in R Before discussing the real numbers, let&s review the basic number systems that precede it, the nat- ural numbers N , the integers Z , and the rational numbers Q : 1.1. The Natural Numbers These are the counting numbers usually denoted by the numerals 1 ; 2 ; 3 ; , and so on. We denote the set of all natural numbers by the symbol N . Intuitively, this is what we get by starting with the "&rst" counting number 1, then successively "adding" 1 to each new number along the way. That is, 2 = 1 + 1 ; 3 = 2 + 1 ; 4 = 3 + 1 ; and so on. This was &rst formalized in the 19th century by Giuseppe Peano. The natural number system N is characterized by the following &ve Peano Axioms; P1 There is a number 1 belonging to N . P2 For each natural number n there is a unique natural number n called the sucessor of n: P3 The number 1 is not the successor of any natural number. P4 If n and m are natural numbers and n = m , then n = m: P5 Suppose A & N ; 1 2 A , and n 2 A whenever n 2 A . Then A = N . Note that axioms P 1 and P 3 together say that 1 is the "&rst" natural number. Axiom P 2 formalizes "adding 1" to each natural number. Axiom P 4 is a prima- tive cancellation law, and axiom P 5 postulates the Principle of Mathematical Induction. To see how we can derive the additive and multiplicative stucture of the natural numbers from the Peano axioms, consider the following de&nitions: De&nition 1. Let n denote a &xed natural number. Then n + 1 = n : and once n + p has been de&ned, let n + ( p ) = ( n + p ) : By P 5 , this de&nes the sum n + m for all natural numbers m . To general the products of all pairs of natural numbers, start o/ again with natural number...
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370_Notes_& solution - Notes for Math 370, Spring 2008...

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