Union of finite and non-regular language Question: ($B$ and $C$ are languages)
$B$ is finite,$C$ isn't regular:

Prove/Disprove: $C\cup B$ isn't regular.

Thoughts:
My intuition says this is true, but I need an idea to prove it. Since I don't know if $C$ as a CFG or RE language I don't know what kind of machine I can build for it.
Hint 1: Try to prove the contrapositive, namely: if $C\cup B$ is regular and $B$ is finite, then $C$ is regular.

Hint 2: Use closure properties of regular languages.