Linear algebra : base, vector space, prove that.

Hi MHF,

I don't know how to prove what I must prove, but I've done something.

Let $\displaystyle \bold{B}$ be a base of the vector space $\displaystyle V$ which has a finite dimension. Let $\displaystyle S \subseteq V$ such that $\displaystyle \bold{B} \subseteq \text{span } (S)$. Prove that $\displaystyle \text{span } (S)=V$.

My *attempt* : We have that $\displaystyle \dim S \leq \dim V$. And also that $\displaystyle \dim \bold{B} \leq \dim S$, but as $\displaystyle \dim \bold{B}=\dim V$ we have $\displaystyle \dim S=\dim V$. So the conclusion follows. Am I right?