To prove that $A \subseteq C$, we rely on the transitivity of the subset relation.

Proof:

1. By definition of subset, $A \subseteq B$ means that for every $x \in A$, $x \in B$.

2. Similarly, $B \subseteq C$ means that for every $x \in B$, $x \in C$.

3. Now, consider any element $x \in A$. Since $A \subseteq B$, it follows that $x \in B$.

4. From $x \in B$ and the fact that $B \subseteq C$, we conclude that $x \in C$.

5. Therefore, for every $x \in A$, $x \in C$.

Conclusion:

This shows that $A \subseteq C$, completing the proof.

Would you like a deeper explanation or a visualization? Here are some related questions to explore:

1. How would the proof differ if $A, B, C$ were proper subsets instead of subsets?
2. Can you give an example of specific sets $A, B,$ and $C$ that satisfy $A \subseteq B$ and $B \subseteq C$?
3. Is this transitivity property valid for other mathematical relations?
4. What happens if $A \not\subseteq B$ but $B \subseteq C$?
5. How is the subset relation different from other set relations, like disjointness?

Tip: Visualize sets using Venn diagrams to better understand subset relationships!