MHB Why is my negation of a math statement incorrect?

[tmt1]

I'm trying to get the negation of this statement:

$\forall C \in \Bbb{R}$ , there is no positive integer $N $ such that if $ k > n$, then $ {x}_{k} > C$I get $\exists C \in R$ such that, $\exists $ a positive integer $N$ , such that if $k > n$ then ${x}_{k} \le C$

but apparently the correct answer is

$\exists C \in R$, $\exists $ a positive integer $N$ , such that if $k > n$ then ${x}_{k} > C$

I can't figure out why

 

[Evgeny.Makarov]

The negation of $\forall C\,\neg A$ is $\exists C\,A$, so the part of the statement after "there is no positive integer $N$" is unchanged.

By the way, the original statement is not stated clearly enough.

$\forall C \in \Bbb{R}$ , there is no positive integer $N $ such that if $ k > n$, then $ {x}_{k} > C$

Are $N$ and $n$ the same? Is $k$ some fixed parameter or is it introduced by another quantifier that is omitted?

 

[tmt1]

The negation of $\forall C\,\neg A$ is $\exists C\,A$, so the part of the statement after "there is no positive integer $N$" is unchanged.

By the way, the original statement is not stated clearly enough.

Are $N$ and $n$ the same? Is $k$ some fixed parameter or is it introduced by another quantifier that is omitted?

I guess my professor is trying to confuse us ...

 

[Evgeny.Makarov]

Presumably, it means, "… integer $N$ such that for all $k$, if $k>N$, …".