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Calculus, ordered set

  1. Sep 4, 2008 #1
    1. The problem statement, all variables and given/known data
    a,b,c,d in a ordered set K and b and d>0
    Show that [tex]\frac{a+c}{b+d}[/tex] stay between the minimum and max from [tex]\frac {a}{b}[/tex] and [tex]\frac {c}{d}[/tex]. Generalize for [tex]a_1,\hdots,a_n,b_1,\hdots,b_n \in K[/tex] with [tex]b_1\hdots,b_n >0[/tex] so [tex]\frac{a_1+\hdots+a_n}{b_1+\hdots+b_n}[/tex] is between the max and min elements from [tex]\frac{a_1}{b_1},\hdots,\frac{a_n}{b_n}[/tex]

    I could do it for the the first case but in a way it's impossible to generalize
    any ideas?
    tks in advance
    Last edited: Sep 4, 2008
  2. jcsd
  3. Sep 5, 2008 #2
    Re: Calculus, field set

    to correct, it`s a ordered field
  4. Sep 15, 2008 #3
    if u consider a/b<c/d
    u can do a/b - (a+c)/(b+d)}=(ad-bc)/(b(b+d))>0
    and c/d - (a+c)/(b+d)}=(bc-ad)/(b(b+d))>0
    and done

    but them using samething for generalizing i couldn't make it :(
    Last edited: Sep 15, 2008
  5. Sep 16, 2008 #4


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    Looks like proof by induction would work.
  6. Sep 16, 2008 #5


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    Let's redo the n=2 case in a way it will be easier to generalize. Write (a1+a2)/(b1+b2)=(b1/(b1+b2))*(a1/b1)+(b2/(b1+b2))*(a2/b2). Notice that the bi/(b1+b2) terms are positive and sum to 1. (This means (a1+a2)/(b1+b2) is in the 'convex hull' of the bi/ai.) If I replace the ai/bi by their minimum and maximum, what do I conclude?
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