Proving that a CFG generates a language
Suppose that we have a context-free grammar $G$ and a set of words $S$, and we would like to prove that the language $L(G)$ generated by $G$ is precisely the set $S$. The most direct way of doing this is to prove $L(G) \subseteq S$ and $S \subseteq L(G)$, which amounts to showing:


For every $x \in L(G)$ we have $x \in S$.
For every $y \in S$ we have $y \in L(G)$.


How this is done depends on how the set $S$ is given, so there is no further, more specific advice that I can offer, except that drawing trees is not really going to result in a proof.