Negation of For all $p/q$ there exists $K$ such that $|\alpha -p/q|\le K/q^n$

Question

What is the negation of the following statement :

For all $p/q$ there exists $K$ such that $|\alpha -p/q|\le K/q^n$.

I'm thinking that For all $p/q$ and for all $K$, |$\alpha -p/q|>K/q^n$.

Am I right ?

Answer 1

Recall that $\neg (\forall x P(x))\equiv \exists x \neg P(x)$ and $\neg (\exists x P(x))\equiv \forall x \neg P(x))$.

Let's do it step by step.

$\neg$ ($\forall p/q$ ($\exists K$ such that $|α−p/q|≤K/q^n$))$\equiv$

$\exists p/q$ such that $\neg$ ( $\exists K$ such that $|α−p/q|≤K/q^n$))$\equiv$

$\exists p/q$ such that $\forall K$ $\neg (|α−p/q|≤K/q^n))\equiv$

$\exists p/q$ such that $\forall K$ $|α−p/q|>K/q^n$.

In english, there exists $p/q$ such that for all $K$, $|α−p/q|>K/q^n$.

Answer 2

Correct me if wrong :

1) There exists a $p/q$ such that for all $K$

the statement $|\alpha - p/q| \le K/q^n$ is not true.

Hence:

2) There exists a $p/q$ such that for all $K$

$|\alpha -p/q| > K/q^n.$