Given the statement, "$\forall x >0, \exists n \in \mathbb{N}$ such that $1/n <x$ and $n > x$".
I concluded that the negation of this statement would be, "$\exists x \leq 0, \forall n \in \mathbb{N}$ such that $1/n \geq x$ or $n \leq x$".
The correct answer is, "$\exists x > 0, \forall n \in \mathbb{N}$ such that $1/n \geq x$ or $n \leq x$"
The difference between my answer and the correct one simply being, "$\exists x > 0$" instead of "$\exists x \leq 0$". Why does the "$>$" not become "$\leq$"?
Best Answer
Let's look at a simpler example. Suppose $A$ is some set of integers, and consider the statement $$\forall x\in A,\text{ $x$ is prime.}$$
In other words, all elements of $A$ are prime.
What's the negation of this? Well, if not all elements of $A$ are prime, that means some number in $A$ is non-prime. Using quantifiers, that's $$\exists x\in A\text{ such that $x$ is not prime.}$$
Notice in particular that the condition $x\in A$ on the quantifier did not get negated: it stayed the same! This makes sense, because we are talking about elements of $A$ the whole time. Numbers that aren't in $A$ are irrelevant to the truth of the original statement, so we don't want to talk about them when forming the negation; instead, we still want to talk only about elements of $A$.
The same thing is going on in your example, just with $x>0$ being the condition on the quantifier instead of $x\in A$.