\lnot (\exists x \in N, x < 3) &\equiv \forall x \in N, \lnot(x < 3)\\
&\equiv \forall x \in N, x \geq 3
Think about what the sentence is saying in words. The original claim is
$N$ contains a number smaller than 3.
Imagine someone says this to you, but you're certain that person is mistaken. What would you have to show them in order to negate the claim? In words, you might say "You are mistaken. The truth is:
$N$ does not contain a number smaller than 3.
In other words:
Every number belonging to $N$ is greater than or equal to 3.
You would most certainly not want to start talking about numbers not belonging to $N$. The person you wish to refute made no claim about such numbers, so neither should your refutation involve them.