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Cheat Sheat 2

Cheat Sheat 2 - limSn= ∞(a Counterexample An=n^2 Bn=-4n(b...

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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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Cheat Sheat 2 - limSn= ∞(a Counterexample An=n^2 Bn=-4n(b...

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