Is a suitable way to prove that any given CFG generates (or not) any given language to draw its total language tree?
What if the tree is infinite? What would then be a better way to prove that a given CFG generates a given language?
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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:
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.
If your language $L$ is, as you state in the comment, given as a regular expression, you can prove that the grammar $G$ generates it in two steps.