I am trying to prove the following claim:
The union of a finite or countable number of sets each of power $c$ is itself of power $c$.
My idea is to use induction, but I cannot finish the proof when there are two sets. Here is what I thought: if $I_1$ and $I_2$ both have continuum cardinality and $I_1\cap I_2=\emptyset$ then it is straightforward to prove. I cannot figure out how to prove if their intersection is non-empty. Any help would be great!