0+x = x

need a proof, that 0+x = x
while x ∈ V and (V, +, •) is a vector space over F

If by 0 you mean the zero vector, then simply,

0 + x
= (0, 0, 0,...) + (x_1, x_2, x_3, ...)
= (0 + x_1, 0 + x_2, 0 + x_3, ...)
= (x_1, x_2, x_3, ...)
= x.

>need a proof, that 0+x = x

1.  Add nothing to everything and you get the same shit you started with.

Qed