Here's the problem:
Prove that the context-free languages are closed under reversal.
Here's my work:
We want to show that if $L$ is a context-free language, then $L^R$ is a context-free language. So let $G$ be the context-free grammar that generates $L$. We construct a new context-free grammar $G'$ as such: for every production $X \rightarrow x$ in $G$, where $X$ is a variable and $x$ is a string of variables or terminals, we add to $G'$ the production $X \rightarrow x^R$. Thus, $G$ generates a word $w$ if and only if $G'$ generates a word $w^R$. Thus, $G'$ generates the language $L^R$, and as $G'$ is a CFG, $L^R$ must be a context-free language.
Is my proof sufficient? Are there any details I should add? Would I want to show "Thus, $G$ generates a word $w$ if and only if $G'$ generates a word $w^R$" by induction?