Every spanning set contains a basis

Statement

Let $F$ be a field, let $V$ be a vector space over $F$, and let $S \subseteq V$ be a spanning set of $V$. Then there is a subset $B \subseteq S$ that is a basis of $V$.

Remarks

For finite $S$, remove from $S$, one at a time, a vector that is a linear combination of the other vectors. When no such vector remains, the set is linearly independent and still spans $V$. In general, take a maximal linearly independent subset of $S$, by Zorn's lemma. For example, $\{(1, 0), (0, 1), (1, 1)\}$ spans $\R^2$ and contains the basis $\{(1, 0), (1, 1)\}$.

Explanations

L1 human

If a set of vectors reaches the whole space, then some part of it is a basis. We throw away vectors that the others can already build. When nothing redundant remains, the remaining vectors form a basis.