Induction of linear independent vectors

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

Let $\displaystyle V$ be a vector-space over a field $\displaystyle \mathbb{F}$; $\displaystyle v_1,...,v_n$ are linear independent vectors in $\displaystyle V$. Prove by induction that the vectors $\displaystyle w_i = \sum_{k=1} ^{i} v_k, 1 \leq i \leq n$, 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:

$\displaystyle \sum_{k=1} ^{i} a_k v_k = 0, 1 \leq i \leq n$, when and only when $\displaystyle a_i = 0$ for all $\displaystyle 1 \leq i \leq k$

So $\displaystyle v_1 = a_1v_1 = 0 = a_1w_1$, but how do you make an induction step out of this? For $\displaystyle w_2$, the proposition is that $\displaystyle v_1 + v_2$ is also linearly independent:

$\displaystyle w_2 = v_1 + v_2 = a_1v_1 + a_2v_2 ; a_1=a_2=0$, 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.