# Does this make sense?

1. Dec 14, 2006

### barksdalemc

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. Dec 14, 2006

### kesh

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
3. Dec 14, 2006

### barksdalemc

yes that is exactly where I am confused, but he is saying that f(p/n) > eps. f(p/n) given 1/n.

4. Dec 14, 2006

### kesh

i'd list all those numbers to exclude them when i'm choosing delta, then any f(x)<1/n<=eps

5. Dec 14, 2006

### HallsofIvy

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.