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Question: 225 show all steps neatly thanks...

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48 Chapter 2. Sequences and Series imit not (c) A divergent sequence such that for every n N it is possible to find n Exercise 2.2.5. Let rll be the greatest integer less than or equal to r. For b) A sequence with an infinite mumber of ones that converges to 1 equal to one. consecutive ones somewhere in the sequence. example, I3 and 1i3] 3. For each sequence, find lim an, and verify it with the definition of convergence. b) a 1(12 + 4n)/3m]l. Refecting on these examples, comment on the statement following Definition 223 that the smaller the e-neighborhood, the larger N may have to be. Exercise 2.2.6. Prove Theorem 22.7. To get started, assume (a)-a nd also that (a)- Now argue e-b Exercise 2.2.7. Here are two usefal delfnitions a set A R if there eists n NEN

2.2. The Limit of a Sequence 43 ries little importance in most cases. The business of analysis is concerned with the We now present what is arguably the most important definition in the book. Definition 2.2.3 (Convergence of a Sequence). A sequence (an) converges lu- at behavior of the infinite tail of a given sequence. is 1g that whenever n 2 N it follows that lan -al an ) --a. The notation limn→oo@n-a is also standard. inequality of this type. to a real number a if, for every positive number e, there exists an NEN such 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 -ase the ending phrase an- a, and think about the points that satisfy an al number a R and a positive number >0.

2.2.5 Show all steps neatly , thanks

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