Prove that if $ L \subseteq \Sigma^\star$ is regular, $L'$ is also regular where

$$ L' = \{w\mid{uw \in L \mbox{ for some string }u \in \Sigma^\star}\}$$

I'm new learning formal language and haven't proved stuff like this. But I'm sharing my thoughts here to make sure I'm on the right track. The first idea came to my mind is using close properties. Since $u$$w$ $\in$ L and $u$ $\in$ $\Sigma^\star$ so we know L is also regular. $L'$ is the concatenation of two regular languages. The regular languages are closed under concatenation thus $L'$ is also regular. 

Any feedback or suggestions?