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Math Help - Positive Sequence with Positive Limit

  1. #1
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    Positive Sequence with Positive Limit

    Suppose that sn is a sequence of positive numbers converging to a positive limit L. Show that there is a c > 0 such that sn > c for all n.


    I can do this for sn < c using the limit definition, but I don't know how to show this.


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  2. #2
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    Re: Positive Sequence with Positive Limit

    Quote Originally Posted by veronicak5678 View Post
    Suppose that sn is a sequence of positive numbers converging to a positive limit L. Show that there is a c > 0 such that sn > c for all n.


    I can do this for sn < c using the limit definition, but I don't know how to show this.


    First of all either learn to use LaTeX or do not use special fonts.

    Suppose that (s_n)\to L>0.
    Using \epsilon=\frac{L}{2} and the definition of sequence convergence prove that there is a p.i. N such that n \geqslant N\, \Rightarrow \,s_n  > \frac{L}{2}.

    Now let c = \min \left\{ {\frac{L}{2},s_1 ,s_2 , \cdots ,s_{N - 1} } \right\} > 0.

    You are done if you fill in the details.
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