Extension of a linearly independent set to a basis

Theorem (Extension to a basis). Let $X$ be a vector space and let $M\subset X$ be a nonempty linearly independent set. Then there exists a basis $B$ of $X$ such that $B\supset M$.

Context. The standard proof uses Zorn’s Lemma (and hence the Axiom of Choice). Combined with the characterization of bases as maximal independent sets (see maximal independence ), it yields the existence of bases in general vector spaces.

Proof sketch. Consider the collection of all linearly independent subsets of $X$ that contain $M$, ordered by inclusion. Any chain has an upper bound given by the union of the chain, which is still independent. Zorn’s Lemma gives a maximal element $B$. Maximality and the cited proposition imply that $B$ is a basis containing $M$.