This is a quick note to prove that two bases of an infinite dimensional vector space have the same cardinality. We freely use the axiom of choice and other standard facts about cardinalities of infinite sets. We will in fact prove the following:

**Theorem**: Let be a vector space with basis with an infinite set. Let be a linearly independent subset of . (e.g. a basis of a subspace). Then .

To prove this, WLOG is a basis of (by extending to a basis of if necessary). For all , write

Let be the set of pairs with . Then has finite fibres, since the sum above is finite, and is surjective, since the lie in the span of the with in the image of , but also the generate . Since is assumed infinite, this is enough to prove that , as required.