Vector space/Finitely generated/Basis/Fact/Proof

Proof

Let ${}v_{i}$, ${}i\in I$, be a finite generating system of ${}V$, with a finite index set ${}I$. We argue with the characterization from

If the family is minimal, then we have a basis. If not, then there exists some ${}k\in I$, such that the remaining family where ${}v_{k}$ is removed, that is ${}v_{i}$, ${}i\in I\setminus \{k\}$, is also a generating system. In this case, we can go on with this smaller index set. With this method, we arrive at a subset ${}J\subseteq I$ such that ${}v_{i}$, ${}i\in J$, is a minimal generating set, hence a basis.

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