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Does this make sense?

  1. Dec 14, 2006 #1
    1. The problem statement, all variables and given/known data
    Consider the Dirichelet's function defined on (0,1) by

    f(x)= 0 is x is irrational
    and
    f(x)= 1/q if x=p/q

    where p and q are positive integers with no common factors. Show that lim f(x) for any x in (0,1) is 0

    3. The attempt at a solution

    Here is the first line of the proof the professor gave us.

    "Let eps>0 and n so large that (1/n)<=eps, where n is a natural number. The only numbers x for which f(x) can > eps are 1/2, 1/3, 2/3, ..., 1/n, ...(n-1)/n. "

    I don't understand how the any p/q with n as q can be greater than epsilon is n is chosen so large that 1/n <=eps.
     
  2. jcsd
  3. Dec 14, 2006 #2
    had trouble reading this last sentence

    do you mean you don't understand how it's possible that p/n > eps given that 1/n <= eps?
     
    Last edited: Dec 14, 2006
  4. Dec 14, 2006 #3
    yes that is exactly where I am confused, but he is saying that f(p/n) > eps. f(p/n) given 1/n.
     
  5. Dec 14, 2006 #4
    i'd list all those numbers to exclude them when i'm choosing delta, then any f(x)<1/n<=eps
     
  6. Dec 14, 2006 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The point is that, within any finite distance of a number a, there can be only a finite number of numerators for a fraction with denominator n: and so only a finite number of fractions with denominator less than or equal to n.
     
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