Here is a proof that was in my notes. My professor did not do a great job of explaining how to prove a set is countable or uncountable so any explanation would be appreciated.

Prove that if A is uncountable and B is any set, then $\displaystyle A\bigcup B$ is uncountable.

I'm thinking I start by saying that A is some set $\displaystyle A={a_{1},a_{2},a_{3}...} $.

Then, I'm thinking maybe you do a proof by cases? Let B be a set defined by $\displaystyle B={b_{1},b_{2},b_{3}...} $. Then assume B is countable. Then, somehow show $\displaystyle A\bigcup B$ is uncountable.

Then assume B is uncountable. Then, somehow show $\displaystyle A\bigcup B$ is uncountable???