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{m+1/n|n is a natural} given m a natural has M as ONLY limit point

  1. Feb 4, 2012 #1
    1. The problem statement, all variables and given/known data
    I wish to show {m+1/n|n is a natural} given m a natural has M as ONLY limit point


    2. Relevant equations

    Rationals dense in reals

    3. The attempt at a solution
    Let M = {m+1/n|n is a natural}.
    I can easily show, by archemidean principle of Reals that m is a limit point of M. I need to show no other real y is a limit point.

    I try to define a set of points K= {j| j is in M and d(j,y) <= d(k,y) for all k in M).

    Essentially if I have that K is nonempty, then since 0<d(y,k) for all k in K, there is q in Q s.t. 0<q<d(y,k) for all k in K, and so a neighborhood of radius q/2 and center at y contains no points of M -> y is NOT a limit point of M.

    THe problem I realized is that m is a number not in M, and so the above situation could be applied to m replacing y in the above, and that I really need to show that the d(j,y) can not be taken arbitrarily small i.e. that K is non empty for y =/= m. I am unsure on how to do this and any help is appreciated.

    Thanks,
     
  2. jcsd
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