Re: Dirichlet function proof

Quote:

Originally Posted by

**alyosha2** for each point

of

.

each open interval of the form

, where

Contains ration

and irrational

.

Ok so far, but then Considering

and

, we have

and

, 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 define

Note that if then . Thus .

Do you see the squeezing now?

Re: Dirichlet function proof

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

Also, why less and not as the sequence can never be more than that value from ?

Re: Dirichlet function proof

Quote:

Originally Posted by

**alyosha2** I'm sorry, I'm still not sure I understand. We have a second sequence

defined on a narrower interval. We are then saying this sequence is squeezed by the interval

on an interval of which the interval of

is a subset. But this only saying that if

then

does, how does this tell us

?

Also, why less

and not

as the sequence can never be more than that value from

?

I wrote it that way out of habit. Usually it is written as

Now if then .

If it is true that then it is true that . The index makes no difference.

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?

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.