36 2.1 Convergence Remark 2.1.3 Note that in the defnition of convergence of a sequence (Defnition 2.1.1), the inequality |an − a| < ε only needs to be satisfed for “suffciently large” values of the index n (n ≥ N). Therefore, it is sometimes convenient to assume that n is already, say, larger than 20, if that helps simplify a calculation. In this case, the ultimate value of N chosen would also need to be greater than 20. The following example illustrates this technique. ■ Example 2.1.5 Consider the sequence given by 2n + 5 an = . 4n2 + n We prove directly from the defnition that {an} converges to 1/4. Let ε > 0 and consider the expression |an − a| given by 2 2 n + 5 1 4n + 20 − 4n2 − n |20 − n| − = = . 4(4n2 + n) 4n2 + n 4 4(4n2 + n) If n ≥ 20, then |20 − n| = n − 20. Therefore, for such n we have |20− n| n− 20 n − 20 n 1 = = ≤ = . 4(4n2 + n) 4(4n2 + n) 4n(4n + 1) 16n2 16n 1 1 Now if n ≥ N, then ≤ 16N . We also observe that 16n 1 1 < ε if N ≥ 16N 16ε . However, we have also required another condition on n, that is n ≥ 20. To account for both 1 conditions, we choose a positive integer N > max 16ε ,20 . Then, for n ≥ N we get 2n + 5 1 n− 20 1 1 − = ≤ ≤ < ε. 4n2 + n 4 4n(4n+ 1) 16n 16N Hence, 2n + 5 1 lim = . n→∞ 4n2 + n 4 Remark 2.1.4 The proof of Theorem 2.1.1 below uses a new technique which is convenient when combining two or more inequalities to achieve an estimate. The defnition of convergence states that for any arbitrary positive number (called ε in Defnition 2.1.1) there exists an index (N) for which a certain inequality is satisfed (|an − a| < ε, for all n ≥ N). Now, if ε > 0, then so is ε/2 (or ε/3, ε/7, 5ε or even cε for any c > 0). So, if an → a there also exists an N1 (possibly different from N) so that |an − a| < ε/2 for all n ≥ N1. This technique will be used several times in subsequent proofs. As anticipated in Defnition 2.1.1, we now prove that a convergent sequence can have only one limit. This justifes referring to the limit of a sequence (rather than to a limit of a sequence). Theorem 2.1.1 A convergent sequence {an} has at most one limit.
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