6.(a) Suppose(an)is increasing. Thenan+1≥anfor alln. Ifk≥0,thenkan+1≥kanfor alln;ifk <0,thenkan+1≤kanfor 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<3.1∈S:s1=√6<3. Thus1∈S.Assume thatk∈S. That is, assumesk<3.Prove thatk+ 1∈S. That is, prove
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