Let be a basis for a vector space and let be a linear independent subset of . Prove there exists a subset of , , such that is a basis for

Proof

Let be the set of all elements in such that is linearly independent. Now consider

Note that every vector in can be expressed as a linear combination of vectors in . So it follows that every vector in can be expressed as a linear combination of vectors in

Thus is linearly independent and generates , so it is a basis.

Is this proof 100% correct?