How do I start this basis proof?

[eyehategod]

Let S be a linearly independent set of vectors from the finite dimensional vector space V.
Prove that there exists a basis for V containing S.

How do I start this proof?
I wasn't able to get it on my test.

 

[Dick]

If S doesn't span V, then pick a vector s1 that's not in the span of S. Let S1={s1}US. Show S1 is linearly independent. If S1 doesn't span V, repeat the process and get S2. Etc. Since dim(V) is finite this has to stop somewhere.

 

[HallsofIvy]

You can also, by the way, prove this for an infinite dimensional vector space but the proof is much harder!