How should I find the negation of this statement?

How should I find the negation of this statement?

I found that I need to find the negation of the following statement when I
tried to give a proof of an exercise:
...For each given $x\in X$ and all $\epsilon>0$, there exists $n>0$ such
that for all $y\in X$, there exists $k\in[0,n]$ such that
$\|f_k(y)-x\|<\epsilon.$
Question 1
I'm confused by the phrase "such that" when I try to rewrite the statement
by symbols:
$\forall x \in X\ \forall\epsilon>0\ \exists n>0\ \forall y\in X\ \exists
k\in[0,n] (\|f_k(y)-x\|<\epsilon)$.
I'm not sure if this is correct or not.
Question 2
Besides, should the negation be
$\exists x \in X\ \exists\epsilon>0\forall n>0\exists y\in X\forall
k\in[0,n](\|f_k(y)-x\|)\geq\epsilon)$?
Question 3
In general, how should one deal with a statement where there are many
"such that"?