Prove this set of vectors is dependant

(v1,v2 ,...,vk ,w) are vectors in vector space V

{ v1 − w,v2 − w,...,vk − w } spans V and w∉Sp{v1,v2 ,...,vk}

prove { v1,v2 ,...,vk } is dependant.

Attempt

I think it has something to do with this:

since { v1 − w,v2 − w,...,vk − w } - containing k vectors - spans V

that means (v1,v2 ,...,vk ,w) - containing k+1 vectors - are dependent

and maybe since w∉Sp{v1,v2 ,...,vk}

that means that in (v1,v2 ,...,vk ,w) w is not a linear combination of (v1,...,vk)

which then means that { v1,v2 ,...,vk } is dependent

What do you guys think??