In fact you are given two languages 

 1. $L_1$ defined as a set of strings of balanced parentheses.  
 2. $L_2$
    defined as a set of strings with equal number of ('s and )'s and
    every prefix of w contains at least as many ('s as )'

You have to prove that these two languages/sets are equal.
One way to prove it is to demonstrate that the grammar you indicated in your post does generate both languages.

Another way is to prove $L_1 \subset L_2$ and $L_2 \subset L_1$.