Is the language given by a context-free grammar always context-free? Consider the language generated by the following grammar:

$S \to aSBb \mid \epsilon$

$B \to aB \mid bB \mid \epsilon$ 

Is the above language context-free?



The above language looks like $\{ w \in (a+b)^* : a^n w_1b w_2b ... w_nb \}$.

I tried using the pumping lemma to show that this language is not context-free, but I was unable to do so.
Hint: A language $L$ is context-free iff there exists a context-free grammar $G$ such that $L = L(G)$. (Some people take this as the definition of context-free languages.)