Induction of linear independent vectors

This is perhaps an easy question- more of a question about mathematical induction than linear algebra, perhaps:

Let be a vector-space over a field ; are linear independent vectors in . Prove by induction that the vectors , are linear independent.

So, the question is asking to prove that the sum of linearly independent vectors is a vector which is linearly independent.

The condition for linear independence is:

, when and only when for all

So , but how do you make an induction step out of this? For , the proposition is that is also linearly independent:

, so you can bracket the scalars out... hmm? But I still can't see how you can make an induction step. I'm sure it's kind of easy, but that's only making it more frustrating.