1. **Problem Statement:** Given two non-empty subsets $S$ and $T$ of a topological space $(X, \tau)$ such that $S \subseteq T$, and a point $p$ which is a limit point of $S$, we need to verify that $p$ is also a limit point of $T$. 2. **Recall the definition of a limit point:** A point $p$ is a limit point of a set $A \subseteq X$ if every open neighborhood $U$ of $p$ contains at least one point of $A$ different from $p$ itself. 3. **Given:** $S \subseteq T$ and $p$ is a limit point of $S$. 4. **To prove:** $p$ is a limit point of $T$. 5. **Proof:** Since $p$ is a limit point of $S$, for every open neighborhood $U$ of $p$, we have: $$ U \cap (S \setminus \{p\}) \neq \emptyset $$ Because $S \subseteq T$, it follows that: $$ U \cap (T \setminus \{p\}) \supseteq U \cap (S \setminus \{p\}) \neq \emptyset $$ Therefore, every open neighborhood $U$ of $p$ contains a point of $T$ different from $p$, which means $p$ is a limit point of $T$. 6. **Conclusion:** We have verified that if $p$ is a limit point of $S$ and $S \subseteq T$, then $p$ is also a limit point of $T$.