Showing $A \subset B \iff A\cap B=A$ Showing 
$A \subset B \iff A\cap B=A$
How would I show this?
My proof
Assume
i. $A \cap B \subset A$
ii.$A \subset A \cap B$
Let $x$ be any element.
Assume $x \in A \cap B$. Then $x \in A$ and $ x \in B$. By hypthesis $x \in A \rightarrow x \in B$ Thus $x \in A$
ii.
Let $ x \in A$ By hypthesis $x \in A \rightarrow x \in B$ 
thus $x \in A \cap B$.
Part 2
$ A\cap B=A \rightarrow A \subset B$
But I find myself stuck here.

*

*If $A \subset B$, then for every $a \in A$, we have $a \in B$ by definition of "$\subset$", thus $a \in A \cap B$. This shows that $A \subset A \cap B$. Now if $a \in A \cap B$ then $a \in A$ by definition of "$\cap$" and thus $A \cap B \subset A$. It follows that $A = A \cap B$.

*If $A \cap B = A$, then for every $a \in A$,  we have $a \in A \cap B$ by definition of "$=$" and thus $a \in B$ by definition of "$\cap$". This shows that $A \subset B$.
