Mathematical Analysis

Download A First Course in Analysis by Donald Yau PDF

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By Donald Yau

This e-book is an introductory textual content on actual research for undergraduate scholars. The prerequisite for this ebook is an exceptional history in freshman calculus in a single variable. The meant viewers of this ebook contains undergraduate arithmetic majors and scholars from different disciplines who use genuine research. given that this publication is geared toward scholars who should not have a lot past adventure with proofs, the velocity is slower in prior chapters than in later chapters. There are hundreds of thousands of routines, and tricks for a few of them are incorporated.

Readership: Undergraduates and graduate scholars in research.

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Example text

But it also has another subsequence {1 − 1, 1 − 12 , 1 − 31 , . }, which converges to 1. 1. 2 Monotone Subsequences Most sequences are not monotone. It may come as a surprise that every sequence has a monotone subsequence. To prove this result, we need the following concept. 7. Give a sequence {an }, a peak is an entry an such that an ≥ ak for all k > n. In other words, a peak is an entry that is also an upper bound for all the entries after it. 7. Let {an } be a sequence. Then it has a monotone subsequence.

2! 3! n! Prove directly from the definition that {an } is a Cauchy sequence. (10) Consider the sequence with an = 1 + 1 1 1 + +⋯+ . 2 3 n Prove that {an } is not a Cauchy sequence, so {an } is divergent. (11) Give an example of a divergent sequence {an } such that lim(an+N − an ) = 0 for every positive integer N . (12) Let a1 = 1, a2 = 2, and an = 21 (an−1 + an−2 ) for n ≥ 3. an = 1 + n−1 (a) For n ≥ 2 prove that an+1 − an = (− 21 ) . (b) Prove that {an } is a Cauchy sequence, hence a convergent sequence.

Prove directly from the definition that {an + bn } and {an bn } are Cauchy sequences. (3) Prove directly from the definition that the following sequences are Cauchy sequences. (a) an = (b) an = (c) an = 1 . 2+3n 2n . 5n−7 3n2 −1 . 2n2 +5 (4) Give an example of a divergent sequence {an } such that lim(an+1 − an ) = 0. (5) Let {an } be a Cauchy sequence such that each an is an integer. Prove that there exists a positive integer N such that {am }m≥N is a constant sequence. (6) Let {an } be a divergent sequence, and let L be a real number.

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