How do you write this in logical symbols: For each $\epsilon>0$ there exist a number $N$ such that $|f_n(x)-f(x)|<\epsilon$ for all $x\in S$ and all $n>N$.
Is this correct? $\forall \epsilon>0 \exists N,(x\in S\land n>N\implies |f_n(x)-f(x)|<\epsilon)$
And how can I negate this?
It’s
$$\forall\epsilon>0\,\exists N\in\Bbb N\,\forall x\in S\,\forall n>N\big(|f_n(x)-f(x)|<\epsilon\big)\;;$$
its negation, after the negation is pulled inside all the quantifiers, is
$$\exists\epsilon>0\,\forall N\in\Bbb N\,\exists x\in S\,\exists n>N\big(|f_n(x)-f(x)|\ge\epsilon\big)\;.$$
This negation can be performed mechanically, using the equivalence of $\neg\forall x\varphi(x)$ with $\exists x\neg\varphi(x)$ and the equivalence of $\neg\exists x\varphi(x)$ with $\forall x\neg\varphi(x)$.
Informally, the negation says that there is a ‘bad’ $\epsilon$, meaning one such that no matter how far out in the sequence $\langle f_n:n\in\Bbb N\rangle$ you go, you can find an $n$ at least that far out and a point $x\in S$ such that $f_n(x)$ and $f(x)$ differ by at least $\epsilon$.