Your proof should divide into two pieces. One part should show that A is a subset of B. The other part should show that B is a subset of A.
Within each part, you should be using the recommended outline for proving a subset inclusion. That is, choose an arbitrary element of the smaller set and show that it belongs to the larger set.
If the main outline of your proof doesn't look like this, rewrite it. Although other methods sometimes work for this kind of problem, the above outline is the standard, recommended method.
Now look at the full solution.