By Donald Yau

This ebook is an introductory textual content on genuine research for undergraduate scholars. The prerequisite for this booklet is an exceptional heritage in freshman calculus in a single variable. The meant viewers of this e-book comprises undergraduate arithmetic majors and scholars from different disciplines who use genuine research. because this e-book is aimed toward scholars who wouldn't have a lot earlier event with proofs, the speed is slower in previous chapters than in later chapters. There are hundreds of thousands of routines, and tricks for a few of them are integrated.

Readership: Undergraduates and graduate scholars in research.

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One direction is in the following result. 9. If {an } is a convergent sequence, then it is a Cauchy sequence. Proof. This is an 2 -argument. Let L be the limit of the sequence. Given there exists a positive integer N such that n ≥ N implies ∣an − L∣ < . 2 Thus, for integers n, m ≥ N , we have ∣an − am ∣ = ∣(an − L) + (L − am )∣ ≤ ∣an − L∣ + ∣L − am ∣ < + = . 2 2 Therefore, {an } is a Cauchy sequence. 5in 48 analysis-yau A First Course in Analysis To show that Cauchy sequences are convergent, we need to work a little harder.

Thus, in the above definition, there is a bijection from S to the set {1, 2, . . , n} if and only if there is a bijection from {1, 2, . . , n} to S. 9. The set N of natural numbers is infinite, which seems obvious. To prove it, suppose to the contrary that there exists a bijection f ∶ {1, 2, . . , n} → N for some positive integer n. We can rank the n natural numbers f (1), . . , f (n) from the smallest to the largest. If f (k) is the largest integer among these n natural numbers, then f (k) + 1 ∈ N is not in the range of f .

9) Let S be the set of functions from Z+ to the set {0, 1} with two elements. Prove that S is uncountable. (10) Let S and T be two sets. Their symmetric difference is defined as the set S △ T = {x ∶ x ∈ S or x ∈ T and x ∈/ S ∩ T }. (a) Prove that S △ T = (S ∖ T ) ∪ (T ∖ S). (b) Prove that S = (S △ T ) ∪ (S ∩ T ), where the union is disjoint. (c) Prove that S ∩ (T △ U ) = (S ∩ T ) △ (S ∩ U ). 5in A First Course in Analysis (d) Prove that (S ∪ T ) △ (U ∪ V ) ⊆ (S △ U ) ∪ (T △ V ). Does equality hold in general?