So I tried doing this problem myself, and the answer that I got seems right, yet at the same time I feel like the way I did it is kind of.... wonky? It seems weird basically, and I was hoping someone can help me validate my answer
Suppose $x \in A \subseteq B$.
Then $x \in A$ and $x \in B$.
Thus, if $x \in A \cap C$,
Then $x\in A$ and $x\in C$ .
Since $x \in A$ and $x \in B$ and $x \in C$,
Then $x \in B$ and $x \in C$,
which implies that $x \in B \cap C$ .
Thus $A\cap C \subseteq B \cap C$ .
Therefore, if $A \subseteq B$, then $A\cap C\subseteq B\cap C$
I'm just not sure if it was alright to assume that $x \in A \cap C$, which is what makes me feel like my proof may be wrong and weird.