# [Math] Cardinality of the union of two infinite set

axiom-of-choiceelementary-set-theory

Suppose that $A$ and $B$ are two infinite sets and $|A|<|B|$. The question is that how to prove that $|A∪B|=|B|$. The proof is related to the Axiom of Choice.

$|A\cup B| \leq |A| + |B| \leq |B| + |B| = |B|$ since $|B|$ is of infinite cardinality and $2\cdot \aleph_k = \aleph _k$ for whichever cardinal $|B|$ happens to be.
Then also $|B|\leq |A\cup B|$ by subadditivity.
Hence $|A\cup B| = |B|$ when $B$ is an infinite set and $|A|\leq |B|$