show that if $|x-\frac{m}{n}| \leq \epsilon$ then $n$ is very large

Question

I am working on my calculus homework currently, and in order to solve a question, I need to prove this more simple statement:

if $|x-\frac{m}{n}| \leq \epsilon$ for all $\epsilon>0$ then $n$ has to be very large.

I understand why this is true, if $n$ is not large, then $x-\frac{m}{n}$ is a large fraction and so it can't be smaller than any epsilon.

But how do I prove it rigorously?

statement to prove: if $\lim_{x_0 \to x} |x-x_0| \leq \epsilon$ and $x_0 = \frac{m}{n} \in \mathbb Q$ then $n$ approaches infinity.

Edit: $x$ is irrational.

Answer 1

Just count. Since $x_0\rightarrow x$ all members of the sequence will be in $[a,b]$ for suitable $a,b$ if the sequence index is large enough. Now there is $m$ such that $a+\frac{m}{n} >b$, so there are at most $m = m(a, b,n)$ fractions with that denominator in $[a,b]$. Similarly for $n-1, n-2, \ldots$. So the number of rationals in $[a, b] $ with denominator less or equal than $n$ is bounded by a constant depending on $a, b, n$.

Answer 2

False!!! Let $n=.00001$. Now show that $m$ just has to be incredibly small.