If L is a regular language, how to prove that L' is also regular? I've been trying to construct a proof of the following statement the whole day but I got stuck:

If $L$ is a regular language, the language $L_{}{'}$ consisting of all words in $L$ containing the letter $\sigma$ (where $\sigma$ is an arbitrary fixed letter in $\Sigma$) is also regular.

I know that the first thing to do is to construct an NFA that recognizes $L_{}{'}$ from the NFA that recognizes $L$, but I can't find a conversion that is general enough. It might me silly, but I think one just have to change the transition function. I hope you guys can give me some advice.
Yes, but changing the transition function also might involve changing the states. In this case, the states should "remember" whether we have seen the letter $\sigma$ in the past.

Alternatively, one can prove closure properties like this using well-known properties, like the fact that regular languages are closed under intersection. The language of all words containing the letter $\sigma$ is regular.