I know what I am trying to prove is very simple, but I am stuck and I need help..

In vector spaces, let V be a vector space over F, c in F and v in V. I am trying to prove the following:

c0 = 0

My proof:

I know the main idea here is to use 0 = v + (-v), but until this point, we don't know that -v = -1.v, we don't know that if v /= 0 then there is an inverse such that v*V^-1 = 1, so let's see..by wa of contradiction, assume c0 /= 0 then

c(v +(-v)) /= 0 I am stuck here..any suggestions?

c0= c(0+0) as usual