Let $A$ and $B$ be sets, then if $A \subseteq B$ then $A \cap B = A$ Let $A$ and $B$ be sets, then if $A \subseteq B$ then $A \cap B = A$.
How do I prove the above? I'm new to proving, please help!
What are the steps I need to create a proof? I'm really lost.
Now I know that the above is true (I can visualize it, and I can even draw a Venn diagram for it) but I don't really follow on formal proofs.
How should I know what to do? Do I use a let $x$ for $A$? or something else?
 A: To prove two sets are equivalent we need to prove that they contain eachother. By definition of the intersection we have $A\cap B\subseteq A$. Now for the opposite inclusion. Choose $x\in A$. Then, since $A\subseteq B$ we have $x\in B$ and thus $x\in A\cap B$, which implies that $A\subseteq A\cap B$. We conclude that $A = A\cap B$.
To get an intuitive feeling for sets I always draw circles on paper and look at the intersections and unions;)
A: Consider what $A\subseteq B$ means: "if $x\in A$, then $x\in B$. Like Marc said in his answer, to show two sets are equivalent, you need to show that they contain each other; that is, in order to show that $A\cap B = A$, then you need to show that $A\cap B\subseteq A$ and $A\subseteq A\cap B$. Let's do this.
$(\subseteq):$ Given $A\subseteq B$. If $x\in A$, then $x\in B$ because $A\subseteq B$; that is, $x\in A\cap B$. Thus, $A\subseteq A\cap B$. 
$(\supseteq):$ Given $A\subseteq B$. If $x\in A\cap B$, then $x\in A$ and $x\in B$. Hence, surely $x\in A$. Thus, $A\cap B\subseteq A$. 
Thus, given $A\subseteq B$, you have shown that $A\cap B\subseteq A$ and $A\subseteq A\cap B$, ultimately showing that $A\cap B = A$, which is what you wanted to show.
