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Riemann Integral - little proof help

  1. Apr 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose f is integrable for all x in[a,b] and f(x)>C ( C is some constant),
    Must show that 1/f is also integrable.


    2. Relevant equations

    f is integrable implies Upf-Lpf<[tex]\epsilon[/tex] for some partition in [a,b]



    3. The attempt at a solution

    Therefore, I must come up with a good [tex]\epsilon[/tex] such that
    Lp(1/f) - Up(1/f) <[tex]\epsilon[/tex]

    Also f is bounded because it's integrable so there must be some m,M such that
    f([a,b])= [m,M]
    in other words f acheives it's minimum and maximum points on the interval.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 16, 2009 #2
    Some hints. For simplicity let's just look at one subinterval [tex]I=[x_i,x_{i+1}][/tex]

    On I, we have [tex]m\le f(x)\le M[/tex] (although really there is no guarantee that m and M are achieved, we still know f is bounded as you said).

    Where does C fit in this inequality?

    Take reciprocals of the inequalities [tex]1/M\le 1/f(x)\le 1/m[/tex] and put 1/c in the correct place, and furthermore where is 0?

    Suppose M-m<delta.

    Then [tex]\frac1m-\frac1M=\frac{M-m}{Mm}[/tex] and you can find an upper bound for the last fraction in terms of delta and C.
     
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