Let $L$ be a context-free language on the alphabet $\Sigma$.
I need to show that for each $a \in \Sigma$ the language $L_a = \{ x \in \Sigma^* \; | \; a.x \in L\}$ is context-free as well.
I wanted to prove it by generating a context-free grammar for $L_a$ by the generation of a new nonterminal for each existing one, to check if an 'a' has been read, but this doesn't work as what I need are all words from $L$ with a leading 'a'.
Could you please help me to find a solution?
Thanks in advance!