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Sequence that has all rational numbers

1. The problem statement, all variables and given/known data

Construct a sequence that has all rational numbers in it

2. Relevant equations

None.

3. The attempt at a solution

Here are my thoughts, though I have no solutions yet.

If I construct a sequence Sn= n*sin(n)-1/n, will it work?

Thanks guys!

 A. Bahat Jan26-12 08:50 AM

Re: Sequence that has all rational numbers

The sine term will often give irrational numbers, so that won't work. Try putting the rationals in an array and finding a path that goes through all of them.

Re: Sequence that has all rational numbers

thanks! can you elaborate a little bit? I'm trying to self-study real analysis, and I'm not really familiar with what you just mentioned...
how would i put them in an array and find a "path"?

thanks!

 SammyS Jan26-12 09:20 AM

Re: Sequence that has all rational numbers

Quote:
 Quote by ask_questions (Post 3728571) 1. The problem statement, all variables and given/known data Construct a sequence that has all rational numbers in it 2. Relevant equations None. 3. The attempt at a solution Here are my thoughts, though I have no solutions yet. If I construct a sequence Sn= n*sin(n)-1/n, will it work? Thanks guys!
Just drop the sine term.

OR

Do you really want a sequence with all of the rational numbers in it. -- maybe just all of the positive rationals?

Better yet: Please type the problem word fro word as it was presented to you.

Re: Sequence that has all rational numbers

Hi guys:

Thank you so much! Here's the problem as it was typed on the book:

Construct a sequence such that every real number is its limit point.

I know this is different from the question I typed above, but my reasoning is that if i can have a sequence that contains all rational numbers, then I can prove that every real number is its limit point. Does that make sense? How should I solve the original question if this does not?

Thank you!

 LCKurtz Jan26-12 12:52 PM

Re: Sequence that has all rational numbers

You might look here for an idea.
http://www.homeschoolmath.net/teachi...-countable.php

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