I'm trying to understand the rules of negating quantifiers. For example the definition of convergence for a null sequence is
$\forall \epsilon > 0, \exists N$ such that $|a_{n}| < \epsilon, \forall n > N$
Now negating, with the help of the comments I get
$\exists \epsilon > 0, ¬ \exists N$ such that $|a_{n}| < \epsilon, \exists n ¬ > N$.
which rewriting it a bit is
$\exists \epsilon > 0 $ such that for all $N, |a_{n}| \geq \epsilon, \exists n \leq N$
A better way to write the original statement is $$\forall \epsilon > 0 \exists N \forall n \gt N, |a_{n}| < \epsilon$$ Then the negation is $$\lnot\forall \epsilon > 0, \exists N \forall n \gt N, |a_{n}| < \epsilon\\ \exists \epsilon > 0,\lnot \exists N \forall n \gt N,|a_{n}| < \epsilon\\ \exists \epsilon > 0,\forall N \lnot\forall n \gt N, |a_{n}| < \epsilon\\ \exists \epsilon > 0,\forall N \exists n \gt N, |a_{n}| \gt \epsilon$$