A Finite Spanning Family Contains a Basis

Let $K$ be a field, let $V$ be a vector space over $K$ with zero vector $0_{V}$, and suppose $V\ne\{0_{V}\}$. Let $n$ be a natural number and let $v\in V^{n}$ be an $n$-tuple in $V$ that spans $V$.

Then there are a natural number $r$ with $r\le n$, in the order on $\mathbb{N}$, and a tuple $e\in V^{r}$ that is a basis of $V$.