Proof by contradiction. Statement negation This should be an easy question. Yet, the provided solution confuses me.
The question comes from "Understanding analysis" by S. Abbot, 2nd edition (Exercise 1.2.11). 

Negate the statement. Make an intuitive guess as to whether the claim or its negation is the true statement.

(b) There exists a real number $x > 0$ such that $x < 1/n\;\;\forall n \in \mathbb{N}$.


The provided solution says:


The solution seems correct, apart from: shouldn't the negation be with $\exists n \in \mathbb{N}$, i.e.:
$$\forall x >0 \;\; \exists n \in \mathbb{N}: x \geq 1/n$$
?



There exists a real number $x > 0$ such that $x < 1/n\;\;\forall n \in \mathbb{N}$.


Is this statement really valid? Let's check.
$nx<1 \;\;\forall n\in\mathbb{N}$ is valid only for $x\leq 0$.
(Because $n$ becomes very large, and if $x\gt 0$ then $nx$ diverges to infinity)
So there is no such $x$ exist. 
