By Patrick M. Fitzpatrick

Simply grasp the basic thoughts of mathematical research with complex CALCULUS. awarded in a transparent and easy manner, this complex caluclus textual content leads you to an exact realizing of the topic via giving you the instruments you want to prevail. a large choice of routines is helping you achieve a real figuring out of the fabric and examples exhibit the importance of what you research. Emphasizing the team spirit of the topic, the textual content indicates that mathematical research isn't really a suite of remoted proof and strategies, yet fairly a coherent physique of information.

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**Example text**

Again, we see that the sequence {sn} is monotonically increasing. We claim that it is not bounded and hence not convergent. Indeed, to see this, observe that 1 1 Sz=1+->1+22 and that 1 3 1 4- 1 4 1 4 2 2 ~=~+-+->~+-+-=1+- 40 1 ADVANCED CALCULUS and 1 Sg 1 1 1 1 3 = S4 + -s + -6 + - + - > S4 + - = 1 + -. 7 82 2 In general, we claim that S2n n > - 1 +2 for every index n. 18) 1, there are 2k-l indices i such that and for each such index, 11 i :=:: 1/2k. Therefore, 2:: 1 1 -;::::2 2k-1 * 1+ - n -. *

14 ADVANCED CALCULUS will argue by contradiction. Suppose there is an integer kin the interval (n, n Then + 1). n < k < n + 1, so that 0 < k - n < 1. As we have already observed, the difference of two integers is again an integer. Thus, k - n is an integer in the interval (0, 1). But we just showed that this is not possible. The assumption that the interval (n, n + 1) contains an integer has led to a contradiction. Thus, the interval (n, n + 1) does not contain any integers. 7 Suppose that S is a nonempty set of integers that is bounded above.

A subsequence of a bounded sequence is bounded. b. A subsequence of a monotone sequence is monotone. c. A subsequence of a convergent sequence is convergent. d. A sequence converges if it has a convergent subsequence. 2. For each of the following statements, determine whether it is true or false and justify your answer. a. Every sequence in the interval (0, 1) has a convergent subsequence. b. Every sequence in the interval (0, 1) has a subsequence that converges to a point in (0, 1). c. Every sequence of rational numbers has a convergent subsequence.