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"?

"Such that" has no mathematical meaning, it simply expresses that the sentence is not over.
And you are right about your translation and the negation.
