By Donald Yau

This publication is an introductory textual content on actual research for undergraduate scholars. The prerequisite for this booklet is an exceptional historical past in freshman calculus in a single variable. The meant viewers of this publication comprises undergraduate arithmetic majors and scholars from different disciplines who use actual research. when you consider that this publication is aimed toward scholars who should not have a lot previous adventure with proofs, the velocity is slower in prior chapters than in later chapters. There are enormous quantities of workouts, and tricks for a few of them are integrated.

Readership: Undergraduates and graduate scholars in research.

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**Extra info for A First Course in Analysis**

**Example text**

3) Consider the subset Q( 2) = {a + b 2 ∶ a, b ∈ Q} of R. Prove that Q( 2) is a field in which the addition and multiplication are the ones in R. (4) Let S be a non-empty bounded subset of R, and let a be a real number. Define the sets a + S = {a + x ∶ x ∈ S} and aS = {ax ∶ x ∈ S}. (a) (b) (c) (d) Prove that a + S and aS are both non-empty and bounded. Prove that inf(a + S) = a + inf(S) and sup(a + S) = a + sup(S). If a > 0, prove that inf(aS) = a inf(S) and sup(aS) = a sup(S). If a < 0, prove that inf(aS) = a sup(S) and sup(aS) = a inf(S).

3 (b) an = (−1)n √ . 3 n+17 √ √ (c) an = n + 17 − n − 5. (5) Let {an } be a monotone sequence. (a) If {an } is not bounded above, prove that an → ∞. (b) If {an } is not bounded below, prove that an → −∞. (6) Let {an } be a bounded sequence. If bn → 0, prove that an bn → 0. (7) Suppose that an → L and bn → M . (a) If an ≤ bn for each n, prove that L ≤ M . (b) Given an example in which an < bn for each n and L = M . (8) Suppose that a ≤ cn ≤ b for each n and that cn → L. Prove that a ≤ L ≤ b. (9) Let {an } be a sequence with an ≥ 0 for each n.

The answer is yes, and the relevant concepts are given in the definitions below. 5. Let {an } be a sequence. (1) The sequence {an } is said to be increasing if an ≤ an+1 for all n. an < an+1 for all n. It is strictly increasing if (2) The sequence {an } is said to be decreasing if an ≥ an+1 for all n. for all n. It is strictly decreasing if an > an+1 (3) The sequence {an } is said to be monotone if it is either increasing or decreasing. It is strictly monotone if it is either strictly increasing or strictly decreasing.