Sequence that has all rational numbers

[ask_questions]

Homework Statement

Construct a sequence that has all rational numbers in it

Homework Equations

None.

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]

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.

 

[ask_questions]

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]

Homework Statement

Construct a sequence that has all rational numbers in it

Homework Equations

None.

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.

 

[ask_questions]

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]

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