Cheat Sheat 2

Cheat Sheat 2 - limSn=+ . (a) Counterexample An=n^2, Bn=...

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Theorem : If (Sn) is an unbounded increasing sequence, then limSn=+ . : Let (Sn) be an increasing sequence and suppose that the set S={Sn:n€N} is unbounded. Since (Sn) is increasing, S is bounded below by S1. Hence S must be unbounded above. Thus, given and M€R, there exists an integer N such that SN>M, so
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Unformatted text preview: limSn=+ . (a) Counterexample An=n^2, Bn= -4n (b) Counterexample An=n-5, Bn=5n 2.36 Remark Since Sn is bounded by the completeness axiom s=sup{Sn,n N}=:S exists, such that Sn <= s for all n. Let eps>0 then Sn <= s +eps. By thm 2.9 |Sn-s|<eps. So limSn=sup(Sn). Simular for infSn....
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This note was uploaded on 05/11/2008 for the course MAT 371 taught by Professor Thieme during the Fall '07 term at ASU.

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Cheat Sheat 2 - limSn=+ . (a) Counterexample An=n^2, Bn=...

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