Question

$\text { Find the limit points of }\left\{\frac{1}{n} \mid n \in \mathbb{N}\right\} \text { in } \mathbb{R} \text {. }$ (DG)

Answer 1

A limit point of a set $A$ in a topological space $X$ is a point $x$ that can be "approximated" by points of $A$ in the sense that every neighbourhood of $x$ with respect to the topology on $X$ also contains a point of $A$ other than $x$ itself.

In our case, for the set $A=\{\frac{1}n∣n∈\N\}$ in $\R$ with standard topoogy the limit point is $x=0.$ Indeed, any basic open neighbourhood $(-\varepsilon,\varepsilon)$ of $x=0$ contains all elements $x_n=\frac{1}n$ for natural numbers $n>\frac{1}{\varepsilon}.$