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Proof with rationals and irrationals

  1. Sep 22, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that any rational in the interval (0,1] can be expressed as a finite sum r=1/q1+1/q2+...+1/qn where the qj are integers and q1<q2<...<qn.

    2. Relevant equations

    3. The attempt at a solution
    Let x[itex]\in[/itex]Q and 0<x[itex]\leq[/itex]1.
    Prove [itex]\exists[/itex]q1, q2, ..., qn[itex]\in[/itex]N with q1<q2<...<qn.

    My professor suggests using the greedy algorithm but I don't understand how that would help the proof.
  2. jcsd
  3. Sep 22, 2011 #2


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    Staff Emeritus
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    Gold Member

    Since you're having trouble tackling the problem for all rationals, have you tried first working on the simpler problem of just considering some rationals? Maybe certain classes of them, or just pick seven at random and see what you can do?
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