TITLE: Help with the proof
QUESTION [0 upvotes]: 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!

REPLY [0 votes]: I don't know, but I suspect this may get at your problem:
Lemma: Let $\mathcal T$ be a set of sets. Then $$\left|\bigcup\mathcal T\right|\le\left|\bigsqcup\mathcal T\right|\le\left|\mathcal T\times\max\bigl\{|T|: T\in\mathcal T\bigr\}\right|,$$ where $\bigsqcup$ represents disjoint union.
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