I have this problem that I cannot figure out. Could you please help me out with this?

Let $r$ be a relation on the set $X$ and let $$R:=\{t:t \text{ is a reflexive relation on } X \text{ with } r\subseteq t \}.$$ For $x,y \in X$ we have $x \, s \, y$ iff for each $t\in R$ we have $x \, t \, y$. Show that $s$ is the reflexive closure of $r$.

My idea so far was as follows.

Suppose $x,y \in X$ and $x \ s \ y$. It follows that for all $t\in R$ we have $x \ t \ y$. Since $r\subseteq t$ for all $t\in R$, it follows $r \subseteq s$ and also $s \subseteq t$. Since each $t\in R$ is reflexive, $s$ is also reflexive.

Could you please verify my idea? I'm sure it has some mistakes.