Prove that if $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$. My attempt, which is pretty scattered and likely incorrect, but just need help collecting ideas and cleaning up the proof.
Direct Proof: Assume $A \subseteq B$ and $B \subseteq C$. Then, $A=B$ and $B=C$, since they are subsets of each other. Therefore, $A=C$, which implies $A \subseteq C$.
 A: Proof is wrong. You need $A\subseteq B$ and $B\subseteq A$ to show that $A=B$. 
Take an element $x\in A$ and see what that implies.
A: The only way to answer this question is to ask yourself at every given point "what does it mean?"
We're given that $A\subseteq B$. What does it mean?
It means that every element in $A$ belongs to $B$, formally for all $x\in A$ we have that $x\in B$.
We're also given that $B\subseteq C$. What does it mean?
It means that every element in $B$ belongs to $C$, formally for all $x\in B$ we have that $x\in C$.
We need to prove that $A\subseteq C$. What does it mean?
It means that given an $x\in A$ we need to show that $x\in C$.
Can you finish now?
A: What you're trying to prove is the transitive relation of subsets. The statement is logically equivalent to if A$\nsubseteq$C  then A$\subseteq$B or B$\nsubseteq$C. So assume A$\nsubseteq$C and B$\subseteq$C, now I must show that A$\nsubseteq$B. So Since A$\nsubseteq$C, $\exists$ x$\in$A such that x$\notin$C, and since B$\subseteq$C and x$\notin$C, it follows that A$\nsubseteq$B.
