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 fact (2). 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, $v_{i}$ , $i \in I ∖ {k}$ , 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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