Negation of a logical statement My question is that when I negate the statement $$(\forall x\in \mathbb{R})( \exists n \in \mathbb{N})(x < 1/n),$$ do I negate all of the statement or just the first part $(\forall x \in \mathbb{R})$?
 A: If I understand correctly, you want to negate the statement
$$\forall x\in \mathbb{R} ~ \exists n \in \mathbb{N} : x < 1/n$$
To do this, you swap every quantifier and then negate the actual formula, i.e. you get
$$\exists x\in \mathbb{R} ~ \forall n \in \mathbb{N} : x \geq 1/n$$
A: You need to negate every block, which gives
$$
(\exists x \in \mathbb{R})(\forall n \in \mathbb{N})(x \geqslant 1/n)
$$
A: It appears you have still not quite "gotten there" just yet--perhaps the following will make the other answers easier to understand and will help as a reference in the future. (More info here if desired)

To negate a statement of the form 
  $$
Q_1x_1 Q_2x_2 \ldots Q_nx_n\; P(x_1,x_2,\ldots,x_n),
$$
  where $Q_i$ is $\forall$ or $\exists$ for $1 \leq i \leq n$, we do the following:
(i) Change every $\forall$ to $\exists$ and every $\exists$ into $\forall$. 
(ii) Replace $P$ by its negation.

In your case, consider the positive "linguistic translation" of what you are dealing with symbolically (of which you are trying to find the negation):

For every real number $x$ there exists a natural number $n$ such that $x<1/n$.

Expressing the above in its more formal symbolic form:
$$
(\forall x\in\mathbb{R})(\exists n\in\mathbb{N})(x<1/n).\tag{1}
$$
How would you negate $(1)$ in light of rules (i) and (ii) specified at the beginning of this answer? First, change every $\forall$ to $\exists$ and every $\exists$ into $\forall$:


*

*$(\forall x\in\mathbb{R})$ becomes $(\exists x\in\mathbb{R})$

*$(\exists n\in\mathbb{N})$ becomes $(\forall n\in\mathbb{N})$


Then replace $P$ by its negation (in this case, we have $P : x<1/n$):
$$
\neg P \equiv x\geq1/n.
$$
Thus, we can finally see that
$$
\neg[(\forall x\in\mathbb{R})(\exists n\in\mathbb{N})(x<1/n)]\equiv(\exists x\in\mathbb{R})(\forall n\in\mathbb{N})(x\geq1/n),
$$
confirming the two other answers you have already received.
