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Math Help - Convergence proof

  1. #1
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    Convergence proof

    Using the given that the sequence {1/n} converges to 0, prove that none of the following assertions are equivalent to the definition of convergence of a sequence {an} to the number a:

    a. For some ε>0 there is an index N such that
    |an-a|<ε for all indices n≥N.

    b. For each ε>0 and each index N
    |an-a|<ε for all indices n≥N.

    c. There is an index N such that for every number ε > 0,
    |an-a|<ε for all indices n≥N.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by uconn711 View Post
    Using the given that the sequence {1/n} converges to 0, prove that none of the following assertions are equivalent to the definition of convergence of a sequence {an} to the number a:

    a. For some ε>0 there is an index N such that
    |an-a|<ε for all indices n≥N.
    Consider the sequence

    a_n=(-1)^n, n=0, 1, ..

    This does not converge.

    but for ε=2 |a_n-0|<ε, for all n>0.

    RonL
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