I'm searching for examples of a transitive relationship R on $\mathbb{R}$ that has the following property:
$\forall(x_n)_n \in \mathbb{R}^{\mathbb{N}}, \forall x, u \in \mathbb{R}, \left\{ \begin{aligned} \forall n \in \mathbb{N}, x_n R u\\ \lim_{n \to \infty} x_n = x \end{aligned} \right. \implies x R u$
An example that leaps to my mind is the following: let $f$ and $g$ be any continuous functions on $\mathbb{R}$. Then the relationship defined by $xRy \iff f(x) \geq g(y)$ (resp., $xRy \iff g(y) \geq f(x)$) satisfies the property. One can think of many other relationships involving continuous functions. Can we construct examples outside this framework?