Every vector space has a basis

Every vector space has a basis.

Proof

We seek a maximal (wrt inclusion) linearly independent subset of .
Then is a basis.

Let partially ordered by inclusion.
Let be a chain in . We show that
is an upper bound for .
We have for all .
We need i.e. is linearly independent.
Assume is a linear relation on .
For each there is some such that
Since is a chain, there is some s.t.
.
Since is linearly independent,
So is linearly independent.
By Zorn’s Lemma, has a maximal element .