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Math Help - Limit Question

  1. #1
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    Limit Question

    How to i prove this?

    If Xn -> X and X > 0, prove that there exists a natural number K such that
    X/2 < Xn < 2X

    for all n in K. (In particular, Xn > 0 for all n in K.)
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by guess View Post
    How to i prove this?

    If Xn -> X and X > 0, prove that there exists a natural number K such that
    X/2 < Xn < 2X

    for all n in K. (In particular, Xn > 0 for all n in K.)
    You observe that for  n large enough that

    |X_n-X|<X/2


    CB
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  3. #3
    MHF Contributor

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    The definition of "limit of a sequence" is " X is the limit of the sequence \{X_n\} if and only if for any ]epsilon> 0, there exist N such that if n> N, then |X_n- X|< \epsilon.

    If X> 0 then so is X/2. Apply the definition of "limit of a sequence" with \epsilon= X/2.

    If |X_n- X|< X/2 then -X/2< X_n- X< X/2 so that X/2< X_n< 3X/2< 2X.
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