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Math Help - visualizing convegence

  1. #1
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    visualizing convegence

    I am trying to get a good visual understanding of convergence so I can better understand the definition of a Cauchy sequence.
    Is the limit aways going to be in the center of the interval, (A-\epsilon, A+\epsilon).
    I guess I dont understand why the \epsilon/2 > 0 is used in the definition.
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  2. #2
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    Re: visualizing convegence

    Quote Originally Posted by delgeezee View Post
    I am trying to get a good visual understanding of convergence so I can better understand the definition of a Cauchy sequence.
    Is the limit aways going to be in the center of the interval, (A-\epsilon, A+\epsilon).
    I guess I dont understand why the \epsilon/2 > 0 is used in the definition.
    I'm not sure what definition you have. To me a sequence $\{x_n\}$ is a Cauchy sequence if

    $\forall \epsilon > 0~~\exists~ m,n \in \mathbb{N} \ni \left|x_n-x_m\right| < \epsilon$

    i.e. the elements get arbitrarily close to one another as the sequence progresses.

    is this any clearer?
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    Re: visualizing convegence

    Im sorry, I meant the proof for theorem that says a sequence is cauchy. It uses the triangle inequality. A sequence is cauchy. iff for each  \epsilon >0 there is a positive integer N s.t. if m,n \geq N, then  \mid a_{n} - a_{m} \mid  > \epsilon} .

    The absolute values make me visualize only half the interval that contain two points in the sequence.
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