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Question: 225 show all steps clearly thanks...

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48 Chapter 2. Sequences and Series (b) A sequence with an infinite number of ones that converges to a limit not equal to one. (c) A divergent sequence such that for every n E N it is possible to find r consecutive ones somewhere in the sequence. Exercise 2.2.5. Let be the greatest integer less than or examp with the definition of convergence. equal to z. For le, r3 and [(3] 3. For each sequence, find lim a, and verify it (a) an = [[5/n]], (b) an (12 +4n)/3n]l. Reflecting on these examples, comment on the st Definition 2.2.3 that the smaller the e-neighborhood, the lar to be. atement following may have Exercise 2.2.6. Prove Theorem 2.2.7, To get started, assume also that (an) b. Now argue a b. Exercise 2.2.7. Here are two useful definitions: an) a and (i) A sequence (an) is eventually in a set A if there exists an N

ces and 43 2.2. The Limit of a Sequence little importance in most cases. The business of analysis is concerned with the se about i behavior of the infinite tail of a given sequence. at progresiWe now present what is arguably the most important definition in the book. rap of letu fs are mer o a real number a if, for every positive number e, there exists an N e N such Definition 2.2.3 (Convergence of a Sequence). A sequence (an) converges that whenever n > N it follows that lan-al < e. stly impro, To indicate that (an) converges to a, we usually write either lim an = a or In an effort to decipher this complicated definition, it helps first to consider the auliug phrase i, -al e, and think about the points that satisfy an ental as (an) → a. The notation limn-wan-a is also standard inequality of this type. 02.2.5 Show all steps clearly thanks

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