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Thread: Dirichlet function proof

  1. #1
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    Dirichlet function proof

    I'm having a hard time understanding this proof.

    for each point $\displaystyle c$ of $\displaystyle R$.

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

    Ok so far, but then

    Considering $\displaystyle \{x_n\}$ and $\displaystyle \{y_n\}$, we have $\displaystyle x_n \to c$ and $\displaystyle y_n\to c$, 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 $\displaystyle c$ of $\displaystyle R$.
    each open interval of the form $\displaystyle (c - \frac{1}{n}, c+ \frac{1}{n})$, where $\displaystyle n \in N$
    Contains ration $\displaystyle x_n$ and irrational $\displaystyle y_n$.
    Ok so far, but then Considering $\displaystyle \{x_n\}$ and $\displaystyle \{y_n\}$, we have $\displaystyle x_n \to c$ and $\displaystyle y_n\to c$, 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 $\displaystyle n\in\mathbb{Z}^+$ define $\displaystyle {I_n}(c) = \left( {c - {n^{ - 1}},c + {n^{ - 1}}} \right)$

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

    Do you see the squeezing now?
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    Re: Dirichlet function proof

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

    Also, why less $\displaystyle < \frac{2}{k}$ and not $\displaystyle < \frac{1}{k}$ as the sequence can never be more than that value from $\displaystyle c$?
    Last edited by alyosha2; May 8th 2013 at 12:59 AM.
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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$\displaystyle x_k$ defined on a narrower interval. We are then saying this sequence is squeezed by the interval $\displaystyle x_n $on an interval of which the interval of $\displaystyle x_k $is a subset. But this only saying that if $\displaystyle x_n \to c$ then $\displaystyle x_k$ does, how does this tell us $\displaystyle x_n \to c$?
    Also, why less $\displaystyle < \frac{2}{k}$ and not $\displaystyle < \frac{1}{k}$ as the sequence can never be more than that value from $\displaystyle c$?
    I wrote it that way out of habit. Usually it is written as $\displaystyle \left| {{x_n} - {y_n}} \right| < \frac{2}{N},~~\forall n\ge N~.$

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

    If it is true that $\displaystyle (x_k)\to c$ then it is true that $\displaystyle (x_n)\to c$. 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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