I started a proof by contradiction and would like to know if I'm on the right track, and get some direction please. I know there are solutions out there for this but I would like to try finishing my proof if possible.

Exercise: Prove that the zero vector is unique. This means that if is any other vector in with the property that for any , then .

My Proof:

By contradiction, assume that the zero vector equals some other vector .

Then by the definition of the zero vector, this means that

(1) .

Since , then for some vector , we can write

(2) , therefore

(3) . By substituting equation (3) into equation (1) we get,

(4) , and thus

(5) .

We have reached a contradiction and therefore proven that the zero vector is unique. QED