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Math Help - limit of an increasing sequence

  1. #1
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    limit of an increasing sequence


    I'm kinda stuck on this problem. Please help me. Thank you much.
    Last edited by john_n82; April 15th 2009 at 11:58 PM.
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  2. #2
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    Quote Originally Posted by john_n82 View Post

    I'm kinda stuck on this problem. Please help me. Thank you much.


    Since x is the supremum of { x_{n}:n\in n},

    GIVEN ε>0 there exists kεN AND SUCH that:


     x-\epsilon< x_{k}\leq x<x+\epsilon

    But since { x_{n}} is increasing and bounded by x:

    for all n x_{n}\leq x and

    for n\geq k x_{n}\geq x_{k}

    hence :


    x-\epsilon<x_{n}\leq x<x+\epsilon\Longleftrightarrow|x_{n}-x|<\epsilon

    THUS \lim_{n\rightarrow\infty}{x_{n}} = x
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  3. #3
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    Thank you for responding. There is a line that I don't understand. Could you please explain further more?

    Quote Originally Posted by xalk View Post
    for n\geq k x_{n}\geq x_{k}
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    Quote Originally Posted by john_n82 View Post
    Thank you for responding. There is a line that I don't understand. Could you please explain further more?
    You know that the sequence is increasing.

    Therefore, x_n  < x_{n + 1}  < x_{n + 2}  <  \cdots  < x_{n + K} .
    So if j>n then x_n < x_j
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