Suppose $R$ is a preorder on $X$. Prove that if $xRy$ then the lower contour set of $y$ is a subset of the lower contour set of $x$, i.e., $\{z \in X: yRz\} \subseteq \{z \in X: xRz\}$.
My working is: Since $R$ is a preorder, it is both reflexive and transitive. So, let $z \in \{z \in X: yRz\}$. Thus, $yRz$ and we know that $xRy$, and since $R$ is transitive, then $xRz$, so $z \in \{z \in X: xRz\}$.
What I don't understand is, no where in the proof did I need the assumption that $R$ is reflexive, so is there something wrong in my proof?
Your proof is correct. The statement is true even if $R$ is only transitive and not reflexive. This is quite common: lots of results you might prove about mathematical structures don't actually require every single axiom that those structures are assumed to satisfy.