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Math Help - sequence problem!

  1. #1
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    Jul 2009
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    sequence problem!

    (a) Suppose that {an} is a convergent sequence of points all in [0, 1] (ie.
    an∈[0, 1] for all n). Prove that limit of an as n-->∞ is also in [0, 1].

    (b) Find a convergent sequence {an} of points all in (0, 1) such that limit of an as n-->∞
    is not in (0, 1).
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  2. #2
    Super Member
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    1) Let \{ a_n \} \subset [0,1] such that a_n \rightarrow a \in \mathbb{R} then for all \epsilon > 0 there exists an N such that if n>N \vert a_n -a \vert < \epsilon then let \vert a \vert \leq \vert a-a_n \vert + \vert a_n \vert < \epsilon + \vert a_n \vert \leq \epsilon + 1 and so \vert a \vert < 1 + \epsilon for all \epsilon >0, and so \vert a \vert \leq 1. In a similar fashion you prove that a can't be negative.

    2) Let a_n =\frac{1}{n}
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