How can there exist a Liouville number?

Question

I just found out about them today so forgive me if I'm just missing something obvious.

As far as I'm aware, the following is true $(\forall a\in\mathbb R\geq0)(((\forall\epsilon\in\mathbb R>0)(a<\epsilon))\Rightarrow (a=0))$.

In the definition of a Liouville number we have $0<|x-\frac p q|<\frac 1 {q^n}$ for any $n$.

Is this not the same as $0<|x-\frac p q|<\epsilon$ for every $\epsilon$? If it is, does that not imply that $|x-\frac p q|=0$ which contradicts the first inequality?

What am I missing?

Answer 1

The definition of a Liouville number is that for every $n>0$ there is some two integers $p,q$ with $q>1$, such that $|x-\frac pq|<\frac1{q^n}$.

You mixed the quantifier on $n$. Since for different $n$ we can use different $p$ and $q$, this is just fine.

Answer 2

You're forgetting about the "there exist $p$ and $q$" part. For any given $n$, you can always find a $p$ and $q$ to make the inequality true. You are thinking about it as if $p$ and $q$ are fixed.