Negate this statement

Question

$\exists \epsilon > 0: \forall\delta >0:\nexists x \in \left\lbrace x: |x-x_0|<\delta \right\rbrace : |x^2_0 - x^2| > \epsilon$

Have been having trouble with this, primarily because it appears really complicated.

Answer 1

$\forall \epsilon > 0: \exists\delta >0:\neg(\nexists x \in \left\lbrace x: |x-x_0|<\delta \right\rbrace : |x^2_0 - x^2| > \epsilon)$$\equiv$ $\forall \epsilon > 0: \exists\delta >0:\exists x \in \left\lbrace x: |x-x_0|<\delta \right\rbrace : |x^2_0 - x^2| \leq \epsilon$

When you take the negation $\forall$ becomes $\exists$ and vice versa. And a statement like $\forall x: P(x)$ becomes $\exists x:\neg P(x)$.