ch12 厦门大学数理经济学.pdfVIP

1 Chapter 12 Limits and Open Sets 1.1 Sequences of real numbers:limit, accumulation point and subsequences Example 12.1 Some examples of a sequence of real numbers a){1, 2, 3, 4, ...} b) 1, 1 , 1 , ... , 2 3 c) 1, 1 , 4, 1 , 16, ... d) 0, − 1 , 2 , −3 , 4 , ... , 2 8 2 3 4 5 e) {−1, 1, −1, 1, −1, ...} f) 2 , 3 , 4 , 5 , ... , 1 2 3 4 g) {3.1, 3.14, 3.141, 3.1415, ...} h) {1, 4, 1, 5, 9, ...} Definition 1 Let {x , x , ...} be a sequence of real numbers and let r be a real number. We say r is 1 2 the limit of this sequence if for any (small) positive number ε, there is a positive integer N such that for all n ≥ N , xn is in the ε - interval about r; that is |xn − r | ε. In this case, we say that the sequence converges to r and we write lim xn = r or lim xn = r or simply xn → r . n→∞ Remark 1 converging sequence need not be distinct from each other; not be from one side; not be monotonic Remark 2 a sequence is said to be bounded if there is a number B such that |xn | ≤ B for all n If the entries of a sequence get arbitrarily close to some number, that number is called an accu- mulation point of the sequence. More formally, r is an accumulation point or cluster point of the sequence {xn } if for any positive ε there are infinitely many elements of the sequence in the interval Iε (r) = {s ∈ R : |s − r | ε}. A limit is a special case of an accumulation