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Can continuum be a countable union of sets with cardinality, less than continuum? I can prove it for a finite union by mathematical induction from this:

$$\mathbb R = A_1 + A_2 \implies |\mathbb R| = |A_1 + A_2| = \max (|A_1| + |A_2|) \implies |A_i| = |\mathbb R|$$

But how to prove it for a countable union?

3 Answers 3