Answer
See the solution.
Work Step by Step
$Proof.$
Let $A$, $B$, and $C$ be sets. Suppose $A$ is a subset of $B$, and suppose $B$ is a subset of $C$. To show $A$ is a subset of $C$, we must show every element of $A$ is an element of $C$. So let $x\in A$. Since $A\subseteq B$, every element of $A$ is an element of $B$. Thus $x\in B$. Since $B \subseteq C$, every element of $B$ is an element of $C$. Hence $x\in C$. Since our choice for $x$ was arbitrary, every element of $A$ is an element of $C$. Therefore, $A\subseteq C._\Box$
