elementary set theory (being a member and subset of a set)? I have to prove or give a counterexample for these two statements:

For the following statements about sets $A$, $B$, and $C$, either prove the statement is true or give a counterexample to show that it is false.
A. If $A \in B$ and $B \subseteq C$, then $A \subseteq C$.
B. If $A \in B$ and $B \subseteq C$, then $A \in C$.

I tried to do it by creating random sets like $A = \{2\}$, $B = \{2,3\}$ and $C = \{2,3,4\}$ and so both statements would be true right? I can't think of a counterexample but I don't know how to actually prove these statements.
Also, if A was the empty set then wouldn't both statements always be true (because the empty set is a member of every other set)?

This first statement is false.
A counter-example could be: $A=\{1\}$, $B=\{\{1\}\}$ and $C=\{\{1\},\{2\}\}$.
Then you have $A\in B$ and $B\subset C$, but you don't have $A\subset C$.
The second statement is true.
To prove it, take $A$, $B$ and $C$ meeting all the conditions. Then $A\in B\subset C$, so you do have $A\in C$ (by the definition of $\subset$). 
