Limit Points in Particular Point Space/Subset/Proof 1

Theorem
Let $T = \struct {S, \tau_p}$ be a particular point space.

Proof
Every open set of $T = \struct {S, \tau_p}$ except $\O$ contains the point $p$ by definition.

So every open set $U \in \tau_p$ such that $x \in U$ contains $p$.

So by definition of the limit point of a set, $x$ is a limit point of $U$.