If $E\subset \mathbb{R}$ is not bounded, then $E$ got no limit points in $\mathbb{R}$. Why?

Question

I have seen a lot of proofs that when a set is not bounded, called $E \subset \mathbb{R}$, then it got no limit points in $\mathbb{R}$. But I am still not convinced.

I was thinking about this set:

$$N = \left\{\frac{1}{n} : n \in \mathbb{N} \right\} \bigcup \left\{n : n \in \mathbb{N} \right\}.$$

Then $N$ is not bounded but $0$ is a limit point of $N$, right? Can someone explain me what am I doing wrong?

Answer 1

You are doing nothing wrong. The statement “when a set is not bounded […] then it got no limit points in $\mathbb R$” is false.