How do I prove that it is an element? I'm trying to prove the transitive property in this proof.   

If $A,B$, and $C$ are sets such that $A \subseteq B$ and  $B \subseteq C$, then $A \subseteq C$.  

How do I show that every element in $A$ is in $C$? The following is what I have so far:   
Let $A, B$, and $C$ are be sets with $A \subseteq B$ and $B \subseteq C$. To show that $A \subseteq C$,
we must show that every element in $A$ is in $C$. To this end we note that if $x \in A$, then
$x \in B$ (because $A \subseteq B$) and therefore $x \in C$ (because $B \subseteq C$). Hence $A \subseteq C$.
 A: 
Let $A, B$, and $C$ are be sets with $A \subseteq B$ and $B \subseteq C$. To show that $A \subseteq C$,
  we must show that every element in $A$ is in $C$. To this end we note that if $x \in A$, then
  $x \in B$ (because $A \subseteq B$) and therefore $x \in C$ (because $B \subseteq C$). Hence $A \subseteq C$.

Your wordsmithing needs refinement, but the logic holds up.
$A, B, C$ are sets such that $A\subseteq B$ and $B\subseteq C$.   To show $A\subseteq C$ we must demonstrate that every element of $A$ is an element of $C$.   We note that because $A\subseteq B$ therefore every element of $A$ is an element of $B$, and because $B\subseteq C$ therefore every element of $B$ is an element of $C$.   We can then conclude that every element of $A$ is an element of $C$; as was to be shown.
$$\begin{align} & (A\subseteq B)\,\wedge\,( B\subseteq C )\\ \forall x:~& (x\in A\to x\in B)\wedge(x\in B\to x\in C)\\ \forall x:~& (x\in A\to x\in C)
\\  & (A\subseteq C) \\\hline \therefore\quad & (A\subseteq B)\,\wedge\,( B\subseteq C )\implies (A\subseteq C)  \end{align}$$
$\require{enclose} \enclose{circle}{\enclose{circle}{\enclose{circle}{\;A\quad}\;B\quad}\;C\quad}$
