Prove that an infinite $(|L|=|\mathbb{N}|) $ language L has an infinite, recognizable and not decidable subset
I have already proved that L has an infinite not recognizable subset A, but I don't know how to do this second part
I have thought about considering B=L\A
And I think it is recognizable because if it weren't then $A \cup B=L$ wouldn't be recognizable either, but I'm not sure about this