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Math Help - Limit Proofs (Sup and Inf)

  1. #1
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    Limit Proofs (Sup and Inf)

    please show and explain
    a. B = lim sup An = {w : w element An for infinite many values of n}
    and C = lim inf An {w:w element An for all but finitely many values of n}


    b. Let (X, F, P) be the probabilty space with X = (0, 1), F the Borel sets and P the uniform distribution. Let An = (1/2 + 1/n, 1 - (1/n))
    Calculate Cn, Bn, lim sup n--> infite An and lim inf n --> infinity An. Does A = lim n--> infitite exist? If so, what is P(A)?


    c. Let An = (0,(1/2) - (1-n)) if n is odd
    ((1/2), 1 - (1-n) if n is even

    again calculate Cn, Bn, lim sup n -->inifinity An and lim inf n-->infinity An. Does lim n --> infinity An exist? If so, what is P(A)?
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  2. #2
    MHF Contributor
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    For a), you need to define the sequence A_n before you can draw any conclusions about it.

    For b) and c), you need to define the sequences B_n and C_n before you can calculate them.
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  3. #3
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    a. The hint given is

    w element of "intersection"(m=n to infinite) "union"(m=n to infinite) Am
    ---> w element "union"(m=n to infinite) Am for all m
    n >N w element of An

    hence how do you you proof this backwards now?


    ohh still not sure about b and c.......
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