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Unformatted text preview: Sequences and Convergence UBC Math 220 Lecture Notes by Philip D Loewen Real Numbers The set R is an ordered field with Q as a subfield and a powerful property called ordercompleteness to be discussed later. Like the rationals, the real numbers enjoy the Archimedean property 1 : for each x in R , there exists a number n in N such that n > x . Sequences Converging to Zero A sequence in R is a function a : N → R . We often write a n for the value a ( n ), and denote the sequence itself by ( a n ) ∞ n =1 . [Notation resembles an “infinite vector”.] The range of the sequence ( a n ) ∞ n =1 is an (unordered) subset of R , namely, ran( a ) = { a n : n ∈ N } . Definition: Let ( a n ) ∞ n =1 be a real sequence. To say “( a n ) ∞ n =1 converges to 0”, or to write a n → as n → ∞ means precisely this: ∀ ε > , ∃ N ∈ N : ∀ n > N,  a n  < ε. ( * ) Example: (a) a n = 0 converges to 0. Pf: Given any ε > 0, pick N = 220. Certainly for every n > 220,  a n  = 0 is less than ε . (b) b n = 1 /n converges to 0. Pf: Given ε > 0, choose any integer N ≥ 1 /ε : then for all n > N ≥ 1 /ε , one has  b n  = 1 /n < ε . (c) c n = sin( nπ/ 2) n converges to 0. Pf: Mimic the proof in (b). Details in class. //// Discussion: Examples (a) and (c) illustrate that some informal descriptions of convergence don’t capture the technical definition. There are sequences that “get closer and closer to zero but never quite touch it”, like example (b), but this is not the only way that a sequence can converge to 0. The definition allows others. The sequences in examples (a) and (c) “touch” 0 infinitely often; also, the sequence in (c) does not get steadily closer to 0, but rather bounces endlessly between touching zero and touching some other value. 1 There are various ways to embed the rational numbers and their rules of arith metic in larger fields. Some of the big fields that result have the Archimedean property and others don’t. The ordercompleteness of the reals is important in explaining why things work out so well for this collection of numbers. File “seq2009”, version of 02 June 2009, page 1. Typeset at 22:10 June 2, 2009. 2 P HILIP D. L OEWEN Set Connection: For each ε > 0, consider a 1 (( ε, ε )) = { n ∈ N : a n ∈ ( ε, ε ) } . [Here we are using relation notation for the relation a 1 in precisely the way defined in our earlier studies.] The definition says that a n → 0 if and only if each set like this contains a subset of the form { N + 1 , N + 2 , N + 3 , . . . } . Key Idea: Error Control: In line ( * ) of the definition above, the number ε > is an “error tolerance”. The latter half of ( * ) says, informally, “For all k sufficiently large, one has a k ∈ ( b a ε, b a + ε ).” The beginning clause of ( * ) shades the whole meaning, adding, “This is true for every ε > 0.” Of course, shrinking ε usually increases the number of outliers....
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 Spring '10
 Loewen
 Real Numbers, Convergence, Limit of a function, Metric space, Limit of a sequence, PHILIP D. LOEWEN

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