Prove that $\mathbb{N}$ has no limit points

Question

How would I prove that $\mathbb{N}$ has no limit points? Why does $\mathbb{N}$ have no limit points?

I have tried proving with integers but clearly this is not the same.

Answer 1

For a point $x$ to be a limit point, we need a sequence {$x_n$} where $x_n \ne x$ in our set to approach $x$.

That does not happen for the set of natural numbers because points are at least one unit apart from each other.

Answer 2

Let us suppose $\mathbb{N}$ has a limit point say $a$. Then, for any $\varepsilon$>0 $\exists$ an open neighborhood $\eta=(a-\varepsilon,a+\varepsilon)$ s.t. $\eta-\{a\}\cap\mathbb{N}\neq\emptyset$. Which is a contradiction as $\mathbb{N}$ contains no points other than integers. So $\mathbb{N}$ has no limit points.