Show that a finite $T_0$-space has at least one point which is a closed subset.
I'm trying to read through the first chapter of this text "Finite Spaces and Larger Contexts" by J. Peter May to get ideas for an undergraduate research project. This is the first exercise he puts to the reader and already I'm a bit lost on this proof (though he does mention it would be challenging for the novice).
Let $x\leq y$ when $x\in \overline{\{y\}}$.
This is a transitive relation on $X$, and $T_0$ axiom means that $x\leq y$ and $y\leq x$ implies $x = y$. So it makes $X$ into a partially ordered set.
The set $\{x\}$ being closed in $X$ means that if $y\leq x$ then $y = x$, that is $x$ is a minimal element of $X$.
But a finite (non-empty) poset has a minimal element.