Let $\mathbb{R}$ form the D1817: Topological ordered set of real numbers.

Let $\lim(E) = \lim_{\mathbb{R}}(E)$ be the D460: Set of limit points of $E \subseteq \mathbb{R}$ in $\mathbb{R}$.

Let $\lim(E) = \lim_{\mathbb{R}}(E)$ be the D460: Set of limit points of $E \subseteq \mathbb{R}$ in $\mathbb{R}$.

Then

(1) | $\forall \, E, F \subseteq \mathbb{R} \, (E \leq F \quad \Rightarrow \quad \lim(E) \leq \lim(F))$ |

(2) | $\forall \, x \in \mathbb{R} : \forall \, E \subseteq \mathbb{R} \, (E \leq x \quad \Rightarrow \quad \lim(E) \leq x)$ |