The first thing you need to do when approaching something like this is to write the definitions down:
- $X\subseteq Y$ if and only if for every $x\in X$ it is true that $x\in Y$.
- $x\in X\cap Y$ if and only if $x\in X$ and $x\in Y$.
Now we assume that $A\subseteq B$ and that $A\subseteq C$, and we wish to show that $A\subseteq (B\cap C)$ as well.
To show that an inclusion holds we take an arbitrary $a\in A$ and we need to show that $a\in B\cap C$. Namely we need to show that $a\in B$ and $a\in C$. Our assumption was that $A\subseteq B$, therefore every element of $A$ is an element of $B$, in particular the $a$ which we took; similarly we assumed $A\subseteq C$ and therefore $a\in C$ as well.
We have therefore proved that if $a\in A$ is any element then $a\in B$ and $a\in C$, and therefore by definition $a\in B\cap C$. Therefore we have shown that the definition of $A\subseteq (B\cap C)$ holds.