3al16_2016.pdf - 1/44 1 Lecture 13 Sequences Finale and...

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1/44 1 Lecture 13 Sequences Finale and Topology of R Sequences Finale Topology of R 2 Lecture 14 Topology of R II 3 Lecture 15 Topology of R III 4 Lecture 16 Topology of R IV Instructor: David Earn Mathematics 3A03 Real Analysis I
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2/44 Thinking About Graduate School? Come find out more at the Grad Info Session! When: Wednesday, October 19, 2016 Time: 4:00 pm – 5:00 pm Where: HH 302 and the Math Café Bartosz Protas and Roman Viveros will talk about graduate programs in Math and Stats, Computational Science and Engineering at McMaster and elsewhere. Miroslav Lovric will also talk about applying to teachers’ college. Current grad students will be present to talk about life in grad school. PIZZA will be served! See you there! Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 13 Sequences Finale and Topology of R 3/44 Mathematics and Statistics Z M d ω = Z M ω Mathematics 3A03 Real Analysis I Instructor: David Earn Lecture 13 Topology of R Wednesday 5 October 2016 Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 13 Sequences Finale 4/44 Sequences Finale! Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 13 Sequences Finale 5/44 Countability Sequences { s n } n =1 are generalizations of the intuitive notion of sets whose elements can be counted. Definition (Countable set) A set is countable if it is finite or is the range a sequence. A set that is not countable is uncountable . Theorem The natural numbers N are countable. (solution on board) Theorem The rational numbers Q are countable. (solution on board) What about R ?. . . Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 13 Sequences Finale 6/44 Countability Theorem (Cantor) The real numbers R are uncountable. (solution on board) Notes : The main argument in the proof is known as “Cantor’s diagonal argument”. We can infer that not only are some real numbers not rational, but there are “many more” real numbers than rational numbers. Cantor’s proof depends on there being a binary expansion for any real number number. . . Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 13 Sequences Finale 7/44 Countability Theorem (Existence and uniqueness of binary expansions) If x [0 , 1) then there is a sequence { a n } such that a n ∈ { 0 , 1 } ∀ n and x = X n = 1 a n 2 n . Specifically, a n = $ x - n - 1 X i = 1 a i 2 i ! 2 n % . Moreover, this binary representation is unique unless x = m / 2 k for some k N and m N , in which case there are exactly two binary representations, the second being given by { b n } where b n = a n n < k , 0 n = k , 1 n > k . Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 13 Sequences Finale 8/44 Countability Theorem (Properties of countable sets) (i) Any subset of a countable set is countable. (ii) The union of a sequence of countable sets is countable.
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