# Dirichlet function proof

• May 7th 2013, 09:37 AM
alyosha2
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?
• May 7th 2013, 10:00 AM
Plato
Re: Dirichlet function proof
Quote:

Originally Posted by alyosha2
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?
• May 8th 2013, 12:57 AM
alyosha2
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 $\displaystyle c$?
• May 8th 2013, 03:20 AM
Plato
Re: Dirichlet function proof
Quote:

Originally Posted by alyosha2
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.
• May 8th 2013, 04:30 AM
alyosha2
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?
• May 8th 2013, 05:33 AM
Plato
Re: Dirichlet function proof
Quote:

Originally Posted by alyosha2
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.