RealAnalysisNotes_Seq_0121.pdf - 2 Sequences A sequence of...

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2 Sequences A sequence of real numbers is a function f : N ! R ; for each counting number n 2 N we associate to it a real number f ( n ) = x n . There are many di erent ways of denoting the sequence. Here are a few which you may see: ( x 1 , x 2 , x 3 , . . . ) = ( x n ) = ( x n ) n 2 N = ( x n : n 2 N ) . The text also writes X = ( x n ), using a single capital letter for the sequence as a whole. Example 2.1. (a) The function f can be explicitly given. For instance, x n = 2 n 2 +1 n 2 , 2 n 2 + 1 n 2 n 2 N = 3 , 9 4 , 19 9 , 33 16 , . . . , . (b) ( x n ) = ( ( - 1) n ) n 2 N = ( - 1 , 1 , - 1 , 1 , - 1 , . . . ) . (c) We may define sequences by iteration. Let g : R ! R be a given real-valued function. Choose an initial value x 1 2 R and then define the sequence iteratively, x n +1 = g ( x n ) , n = 1 , 2 , 3 , . . . This is a natural way to define sequences, for example to approximate solutions to equa- tions. For a more specific example, take g ( x ) = 1 2 ( x + 2 x ) , and generate the sequence ( x n ) by iteration: x 1 = 2 and x n +1 = 1 2 x n + 2 x n , n = 1 , 2 , 3 , . . . . The first few values are: ( x n ) = (2 , 1 . 5 , 1 . 41 ¯ 6 , 1 . 414215686 . . . , 1 . 414213562 . . . , . . . ) Later, we will prove that this sequence converges to p 2 . Note that a sequence is not the same thing as a set. A sequence is an infinite ordered list of numbers. In a sequence the same number may appear several times, and changing the order of the elements of a sequence creates an entirely di erent sequence. A set has no order, and there is no point in repeating the same value several times. Taking Example (b) above, { x n : n 2 N } = { - 1 , +1 } is a set with two elements. The sequence ( x n ) is not the same thing. If we let { y n } n 2 N = ( - 1 , - 1 , 1 , 1 , - 1 , - 1 , 1 , 1 , . . . ), this is a di erent sequence than { x n } n 2 N , yet it takes values in the same set { y n : n 2 N } = { - 1 , +1 } . 12
2.1 Limits of Sequences Definition 2.2. We say that the sequence ( x n ) converges to x 2 R if: ( for every " > 0 there exists K 2 N so that | x n - x | < " for every n K . We write x = lim n !1 x n , or x n ---! n !1 x as n ! 1 . If there is no value of x for which x n converges to x , we say ( x n ) diverges . Example 2.1 (a) converges to x = 2, x n = 2 n 2 +1 n 2 ---! n !1 2 as n ! 1 . Let’s verify this via the definition: let " > 0 be given. We calculate | x n - 2 | = 2 n 2 + 1 n 2 - 2 = (2 n 2 + 1) - 2 n 2 n 2 = 1 n 2 = 1 n 2 . (2.1) We need to determine when the right-hand side 1 /n 2 < " . This is true when n > 1 / p " . By the Archimedean property, we may choose K 2 N with K > 1 / p " , so for all n K , n K > 1 / p " and so 1 n 2 < " for n K . Plugging back into (2.1) we have | x n - 2 | = 1 n 2 < " 8 n K, and so x n ---! n !1 2 by definition. Remark 2.3. By a Practice Problem, the inequality condition | x n - x | < " is equivalent to two inequalities, above and below, x - " < x n < x + " . Another more geometrical way to write this is in terms of open intervals, x n 2 ( x - " , x + " ) . The interval ( x - " , x + " ) is called an open neighborhood of size " centered at x . So x n ---! n !1 x is equivalent to saying that the elements of the sequence x n 2 ( x - " , x + " ) eventually always , that is, for all n N ( " ). We verify the following properties of the limit for sequences: Theorem 2.4. Assume the sequence ( x n ) n 2 N is convergent, x = lim n !1 x n .

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