Suppose A and B are sets. Prove that $A\subseteq B$ if and only if $A \cap B = A$. Suppose $A$ and $B$ are sets. Prove that $A \subseteq B$ if and only if $A \cap B = A$.

Here's how I see it being proved.
If $A$ and $B$ are sets,and the intersection of $A$ and $B$ is equal to $A$, then the elements in $A$ are in both the set $A$ and $B$. Therefore, the set of $A$ is a subset of $B$ since all the elements are contained in the interesection of sets $A$ and $B$ are equal to $A$.
Can I prove it that way?
Your proof is almost perfect and let me rectify it a bit:

Let  $A$ and $B$ be two sets. The intersection of $A$ and $B$ is equal to $A$, is equivalent to  the elements in $A$ are in both the set $A$ and $B$ which's also equivalent to the set of $A$ is a subset of $B$ since all the elements of $A$ are contained in the intersection of sets $A$ and $B$ are equal to $A$.
Suppose A is subset of B. Let X belongs to A then by hypothesis, X will belong to B. Hence X belong to A  and 
X  belong to B  implies that X belongs to A intersection B.  Accordingly A is subset of A intersection B. But we know that A intersection B is always subset of A. Hence A intersection B is equal to A. 
On the other hand, suppose A intersection B is equal to A. Then in particular, A is subset  of A intersection B. We know that, A intersection B is subset of B. Hence A is subset  of B.