8 CHAPTER 1. LIMIT 日 日日 Figure 1. 1.1: Sequ Definition 1.1.1 (Intuitive). If In approaches a finite number I when n gets bigge and bigger, then we say that the sequence n converges to the limit l and write lim A sequence diverges if it does not approach a specific finite number when n gets The sequences n, Zn, Un converge respectively to 2, 0 and T. The sequences n, un, Un diverge. Since the limit describes the behavior when n gets very big, we have the following propert Proposition 1. 2. If yn is obtained from Tn by adding, deleting, or changing finitely many terms, then lim→oxn=limn→yn The equality in the proposition means that In converges if and only if yn con verges. Moreover, the two limits have equal value when both converge ample 1.. The sequence Vn+2 is obtained from Vn by deleting the first two terms. By limn→sJn =0 and Proposition 1. 1.2, we get lim, n+2 In general, we have limn-oo n+k= limn-yoo n for any integer k The example assumes limn-+oo 0, which is supposed to be intuitively obv ous. Although mathematics is inspired by intuition, a critical feature of mathematics is rigorous logic. This means that we need to be clear what basic facts are assumed in any argument. For the moment, we will always assume that we already know8 CHAPTER 1. LIMIT n xn yn zn un vn wn Figure 1.1.1: Sequences. Definition 1.1.1 (Intuitive). If xn approaches a finite number l when n gets bigger and bigger, then we say that the sequence xn converges to the limit l and write limn→∞ xn = l. A sequence diverges if it does not approach a specific finite number when n gets bigger. The sequences yn, zn, wn converge respectively to 2, 0 and π. The sequences xn, un, vn diverge. Since the limit describes the behavior when n gets very big, we have the following property. Proposition 1.1.2. If yn is obtained from xn by adding, deleting, or changing finitely many terms, then limn→∞ xn = limn→∞ yn. The equality in the proposition means that xn converges if and only if yn con￾verges. Moreover, the two limits have equal value when both converge. Example 1.1.1. The sequence 1 √ n + 2 is obtained from 1 √ n by deleting the first two terms. By limn→∞ 1 √ n = 0 and Proposition 1.1.2, we get limn→∞ 1 √ n = limn→∞ 1 √ n + 2 = 0. In general, we have limn→∞ xn+k = limn→∞ xn for any integer k. The example assumes limn→∞ 1 √ n = 0, which is supposed to be intuitively obvi￾ous. Although mathematics is inspired by intuition, a critical feature of mathematics is rigorous logic. This means that we need to be clear what basic facts are assumed in any argument. For the moment, we will always assume that we already know