Can Someone please SOLVE and EXPLAIN this problem Thank You

Can Someone please SOLVE and EXPLAIN this problem. Thank You in Advance

let V be an arbitary vector space over R. Prove that: 0.V = 0 vector.

in words, (the Scaler 0)(dot product)(Any Vector in V) = (Zero Vector in V).

Solution

Let us look at element by element multiplication by 0

Case 1. V=0

there is nothing to prove

Case 2. V is non zero

So let v be a non zero vector in V

We know 0+0=0

So,

0.v=(0+0).v=0.v+0.v

Let, 0.v=w

w=w+w

Adding -w to both sidse gives

w-w=w

Hence, w=0