Let be a field whose cardinality is countable (that is, finite or countably infinite). Let be an infinite dimensional vector space over having basis . prove that and have the same cardinality.
Note that . Certainly we have , since .
Let be an infinite set. Given a natural number , denote by the set of all subsets of having cardinality . Note that , since is infinite. Denote by the set of all finite subsets of ; we also have . (This sum denotes a cardinal sum.) Thus the set of all finite subsets of has the same cardinality as .
Now every element of is uniquely an -linear combination of finitely many elements of . Thus we have .
Thus we have .