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Math Help - Dirichlet function proof

  1. #1
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    Dirichlet function proof

    I'm having a hard time understanding this proof.

    for each point c of R.

    each open interval of the form (c - \frac{1}{n}, c+ \frac{1}{n}), where n \in N
    Contains ration x_n and irrational y_n.

    Ok so far, but then

    Considering \{x_n\} and \{y_n\}, we have x_n \to c and y_n\to c, by the squeeze rule.

    I don't understand what's going on here. As there are only two sequences, how are these two sequences being squeezed?
    Last edited by alyosha2; May 7th 2013 at 09:47 AM.
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  2. #2
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    Re: Dirichlet function proof

    Quote Originally Posted by alyosha2 View Post
    for each point c of R.
    each open interval of the form (c - \frac{1}{n}, c+ \frac{1}{n}), where n \in N
    Contains ration x_n and irrational y_n.
    Ok so far, but then Considering \{x_n\} and \{y_n\}, we have x_n \to c and y_n\to c, by the squeeze rule. I don't understand what's going on here. As there are only two sequences, how are these two sequences being squeezed?
    EACH sequence is being squeezed separately.

    If n\in\mathbb{Z}^+ define {I_n}(c) = \left( {c - {n^{ - 1}},c + {n^{ - 1}}} \right)

    Note that if k\ge N then {I_k}(c)\subseteq{I_N}(c). Thus \left| {{x_k} - c} \right| < \frac{2}{k} \leqslant \frac{2}{N}.

    Do you see the squeezing now?
    Thanks from topsquark
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  3. #3
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    Re: Dirichlet function proof

    I'm sorry, I'm still not sure I understand. We have a second sequence  x_k defined on a narrower interval. We are then saying this sequence is squeezed by the interval x_n on an interval of which the interval of x_k is a subset. But this only saying that if  x_n \to c then x_k does, how does this tell us x_n \to c?

    Also, why less < \frac{2}{k} and not < \frac{1}{k} as the sequence can never be more than that value from c?
    Last edited by alyosha2; May 8th 2013 at 12:59 AM.
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  4. #4
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    Re: Dirichlet function proof

    Quote Originally Posted by alyosha2 View Post
    I'm sorry, I'm still not sure I understand. We have a second sequence  x_k defined on a narrower interval. We are then saying this sequence is squeezed by the interval x_n on an interval of which the interval of x_k is a subset. But this only saying that if  x_n \to c then x_k does, how does this tell us x_n \to c?
    Also, why less < \frac{2}{k} and not < \frac{1}{k} as the sequence can never be more than that value from c?
    I wrote it that way out of habit. Usually it is written as \left| {{x_n} - {y_n}} \right| < \frac{2}{N},~~\forall n\ge N~.

    Now if |x_n-c|<\frac{1}{N} then |x_n-c|<\frac{2}{N}.

    If it is true that (x_k)\to c then it is true that (x_n)\to c. The index makes no difference.
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    Re: Dirichlet function proof

    I'm not sure I see. But here 's my understanding so far. Each rational/irrational number in each interval we define gets closer to the point on which we have defined the interval the narrower we make the interval. So a sequence on the larger interval will squeeze the sequence on the smaller interval. But I'm not sure how this helps. Are we saying something like we are creating a sequence of intervals and then extracting numbers from within those intervals to create a sequence of numbers that tend to the point around which the interval is defined?
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    Re: Dirichlet function proof

    Quote Originally Posted by alyosha2 View Post
    Are we saying something like we are creating a sequence of intervals and then extracting numbers from within those intervals to create a sequence of numbers that tend to the point around which the interval is defined?
    That is correct.
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