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3al11_2016.pdf - 1/50 1 Lecture 6 Notion of Distance or...

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1/50 1 Lecture 6 Notion of Distance or Metric Sequences 2 Lecture 7 Sequences II 3 Lecture 8 Sequences III 4 Lecture 9 Sequences IV 5 Lecture 10 Sequences V 6 Lecture 11 Sequences VI Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 6 2/50 Mathematics and Statistics Z M d ω = Z M ω Mathematics 3A03 Real Analysis I Instructor: David Earn Lecture 6 Sequences Monday 19 September 2016 Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 6 3/50 Announcements Niky Hristov office hours: Wed and Thurs, 12:30–1:30pm (Hamilton Hall 403) Steven Lazzaro at Math Help Centre: Tuesday 2:30–6:30pm Solutions to Assignment 1 were posted on Friday evening. Study them! Remember that Dr. Valeriote’s 2015 3A03 assignments and tests (together with solutions) are available at http://ms.mcmaster.ca/ ~ matt/3a3.html . Take advantage of these problems and solutions from a previous version of this course. They provide many useful examples that should help you prepare for tests and the final exam. (Note that while most of the content of the course is the same this year, there are some differences.) Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 6 4/50 Putnam Competition The William Lowell Putnam competition is a university-level mathematics competition held annually for undergraduate students at North American universities. More information can be found at http://www.math.mcmaster.ca/undergraduate Follow the Putnam competition link under “Useful Links” at the bottom of the page. This year’s competition will occur on Saturday Dec. 3. If you are interested in participating or learning more, send email to David Earn, [email protected] or Bradd Hart, [email protected] . In your e-mail please state what program and year you are in. There will be an information session on Wednesday, Sept. 21 at 11:30am in HH-410. Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 6 5/50 Question Did you read the posted solutions to Assignment 1? Any questions about them? Office hour TODAY after class (HH-317). Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 6 Notion of Distance or Metric 6/50 The metric structure of R ( § 1.10) Definition (Distance function or metric) The distance between two real numbers x and y is d ( x , y ) = | x - y | . Theorem (Properties of distance function or metric) 1 d ( x , y ) 0 distances are positive or zero 2 d ( x , y ) = 0 ⇐⇒ x = y distinct points have distance > 0 3 d ( x , y ) = d ( y , x ) distance is symmetric 4 d ( x , y ) d ( x , z ) + d ( z , y ) the triangle inequality Note : Any function satisfying these properties can be considered a “distance” or “metric”. Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 6 Notion of Distance or Metric 7/50 The metric structure of R ( § 1.10) Given d ( x , y ) = | x - y | , the properties of the distance function are equivalent to: Theorem (Metric properties of the absolute value function) For all x , y R : 1 | x | ≥ 0 2 | x | = 0 ⇐⇒ x = 0 3 | x | = |- x | 4 | x + y | ≤ | x | + | y | ( the triangle inequality ) Instructor: David Earn Mathematics 3A03 Real Analysis I
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Lecture 6 Sequences 8/50 Sequences A sequence is a list that goes on forever.
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