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Homework Help: Sequence that has all rational numbers

  1. Jan 26, 2012 #1
    1. The problem statement, all variables and given/known data

    Construct a sequence that has all rational numbers in it

    2. Relevant equations


    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!
  2. jcsd
  3. Jan 26, 2012 #2
    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.
  4. Jan 26, 2012 #3
    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"?

  5. Jan 26, 2012 #4


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    Just drop the sine term.


    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.
  6. Jan 26, 2012 #5
    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!
  7. Jan 26, 2012 #6


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