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Bounded sets

  1. Jul 18, 2007 #1

    daniel_i_l

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    Gold Member

    1. The problem statement, all variables and given/known data
    f(x) = (x+1)/x^2
    a)prove that f is bounded in N (N is the set of natural numbers so we have to prove that f(N) is a bounded set)
    b)find supf(N) and inff(N).
    c) does f have a maximum or minimum in N?


    2. Relevant equations



    3. The attempt at a solution
    First I proved that for every x,y >= 1, if x<y then f(x)>f(y):
    y>x>=1 and so y^2 > x^2 and so y^2 - x^2 > 0
    xy=yx and so x^2 * y < y^2 * x and so
    y^2 * x - x^2 * y > 0 and together
    y^2 * x - x^2 * y + y^2 - x^2 = y^2 * (x+1) - x^2 * (y+1) > 0 and so
    (y^2 * (x+1) - x^2 * (y+1)) / (x^2 * y^2) = (x+1)/(x^2) - (y+1)/(y^2) > 0and so (x+1)/(x^2) > (y+1)/(y^2).

    Now, f(1) = 2 and so for all x>1 f(x)<2 and so maxf(N) = supf(N) = 2.

    Also, for every x>=1 f(x)>0. The limit of f at infinity is 0. So if f(N) has a lower bound c>0 then since f has a limit of zero at infinity we can find some M>0 so that for every x>M (we can find an x in N) |f(x)|<c => f(x)<c which means that c isn't a lower bound so inff(N) = 0 and there's no minimum.

    Is that right (especially the proof)? Does it matter that I did a,b and c in the same step?
    Thanks.
     
  2. jcsd
  3. Jul 18, 2007 #2
    Instead of going 'since xy=yx, x^2y<y^2x' is do this: 'since x<y, multiplying both sides by xy give x^2y<y^2x'. The way you originally put it was a head scratcher.

    You haven't really done a,b,c in one step, what you've shown is that the function [itex]f:R_{\ge 1} \to R[/itex] is monotonically decreasing. Then you considered the restriction to natural numbers and then made your arguments based on that.

    If you want to break it up into parts and be more organized then you can first do the preliminary work of showing that [itex]f:R_{\ge 1} \to R[/itex] is monotonically decreasing and then do a,b, and c in that order. You haven't explicitly claimed why f(N) is bounded.

    The majority of the work is correct. You could be more liberal with your explanations, but not necessary. As an example, the supremum exists because the maximum exists. I don't know if that's what you meant because you just wrote it as maxf(N)=supf(N)=2 when your work only showed that f(N) has a maximum of 2.

    Just little things that your professor might pick on.
     
    Last edited: Jul 18, 2007
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