For large n s n s and s n 1 s Therefore s 1 4 2s 7 4 s 2 s 7 s 7 2 4a 1 n

For large n s n s and s n 1 s therefore s 1 4 2s 7 4

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6.(a) Suppose(an)is increasing. Thenan+1anfor alln. Ifk0,thenkan+1kanfor alln;ifk <0,thenkan+1kanfor alln. In eithercase,(kan)is monotone.(b) Counterexample: Let (an) = (1,2,2,3,3,4,4, . . .) and let (bn) = (1,1,2,2,3,3,4,4, . . .).(an)and(bn)are increasing sequences. The sequencean/bnis:parenleftbigg1,2,1,32,1,43, . . .parenrightbiggwhich is not monotone.7. It is clear thatsnis increasing.Claimsn<3for alln.By induction: LetSbe the set of positive integersfor whichsn<3.1S:s1=6<3. Thus1S.Assume thatkS. That is, assumesk<3.Prove thatk+ 1S. That is, prove 5
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  • Fall '08
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  • Math, lim, Natural number, Sn

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