Finitely Generated Vector Space has Finite Basis

Theorem
Let $K$ be a division ring.

Let $V$ be a finitely generated vector space over $K$.

Then $V$ has a finite basis.

Proof
This follows from Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set.

It suffices to find:
 * A linearly independent subset $L\subset V$
 * A finite generator $S\subset V$

with $L\subset S$.

By Empty Set is Linearly Independent, we make take $L=\varnothing$ and $S$ any finite generator, which exists because $V$ is finitely generated.

Also see

 * Vector Space has Basis, which does not assume that $V$ is finitely generated