For a discrete metric space in $\mathbb{R}$, $0$ is not a limit point for $\left\{\frac{1}{n}:n\in\mathbb{N}\right\}$.

Question

Statement: For a discrete metric space in $\mathbb{R}$, $0$ is not a limit point for $\left\{\frac{1}{n}:n\in\mathbb{N}\right\}$.

I have difficulty to understand why $0$ is not a limit point.

Answer 1

Because the open ball centered at $0$ with radius $1$ contains no number of the form $\frac1n$ ($n\in\mathbb N$).

Answer 2

There exists one open neighbourhood of 0 that doesn’t contain any point of $\{1/n\vert n=1,2,...\}$. Take the open set $\{0\}$ which is open in discrete topology of $\mathbb R$ and you are done.

Answer 3

In the discrete metric, there is no sense of "close." Either numbers are equal or they are not.