Finite Elements in a Set of Rational Numbers Proof

[RPierre]

Homework Statement

This problem is insanely intuitive.

Define f : (0,1) \rightarrow \Re by

f(x)=\begin{cases} <br /> 1/q&amp;\text{if } x \neq 0 \text{, is rational, and }x = p/q \text{in lowest terms}\\ <br /> 0&amp;\text{otherwise }\end{cases}

Suppose \epsilon &gt; 0. Prove that there are at most a finite number of elements y\in(0,1) such that f(y)\geq\epsilon

Homework Equations

Must be a rigorous proof. Thats about it.

The Attempt at a Solution

I have no Idea where to START in solving this. All that I know is over (0,1), f(y) is always going to be 1/q, and therefore y must be rational in the form p/q. Otherwise, I have nothing on this one.

 

[Office_Shredder]

f(y)=1/q only if y=p/q. If y is irrational which it surely can be between 0 and 1, then f(y)=0.

This is pretty straightforward. Answer the question: For what values of q is 1/q >= epsilon? So what does that tell you about the values of y such that f(y) >= epsilon?