TITLE: The cardinality of a countable union of sets with less than continuum cardinality
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: As remarked by others, assuming the axiom of choice this is false.
However there are several interesting cases when the axiom of choice fails:

The real numbers may be a countable unions of countable sets.
The real numbers may be the union of two sets both of cardinality strictly less than the continuum, furthermore at least one of these sets can be split into two smaller sets, so we can even split the continuum into three sets of smaller cardinality; and so on.