Let $\mathbb{R}$ be the D1817: Topological ordered set of real numbers such that

(i)	$x, y : \mathbb{N} \to \mathbb{R}$ are each a D336: Convergent sequence in $\mathbb{R}$
(i)	\begin{equation} \forall \, n \in \mathbb{N} : x_n \leq y_n \end{equation}

Then \begin{equation} \lim_{n \to \infty} x_n \leq \lim_{n \to \infty} y_n \end{equation}

Let $\mathbb{R}$ be the D1817: Topological ordered set of real numbers such that

(i)	$x, y : \mathbb{N} \to \mathbb{R}$ are each a D336: Convergent sequence in $\mathbb{R}$

Then \begin{equation} x \leq y \quad \implies \quad \lim_{n \to \infty} x_n \leq \lim_{n \to \infty} y_n \end{equation}