Generalizing the Replacement Theorem

Let $\beta$ be a basis for a vector space $V$ and let $S$ be a linear independent subset of $V$ . Prove there exists a subset of $\beta$ , $S_1$ , such that $S\cup{S_1}$ is a basis for $V$

Proof
Let $S_1$ be the set of all elements in $\beta$ such that $S\union{S_1}$ is linearly independent. Now consider $S\cup{S_1}$

Note that every vector in $S$ can be expressed as a linear combination of vectors in $\beta$ . So it follows that every vector in $V$ can be expressed as a linear combination of vectors in $S\cup{S_1}$
Thus $S\cup{S_1}$ is linearly independent and generates $V$ , so it is a basis.

Is this proof 100% correct?