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 $\displaystyle \vec{z}$ is any other vector in $\displaystyle \mathbb{R}$ with the property that $\displaystyle \vec{z}+\vec{v}=\vec{v}$ for any $\displaystyle \vec{v}\in\mathbb{R}^{n}$, then $\displaystyle \vec{z} = \vec{0}_{n}$.

My Proof:

By contradiction, assume that the zero vector equals some other vector $\displaystyle \vec{z}\in\mathbb{R}^{n}$.

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

(1) $\displaystyle \vec{0} + \vec{z} = \vec{0}$.

Since $\displaystyle \vec{0}=\vec{z}$, then for some vector $\displaystyle \vec{v}\in\mathbb{R}^{n}-\vec{0}$, we can write

(2) $\displaystyle \vec{z}+\vec{v}=\vec{0}$, therefore

(3) $\displaystyle \vec{z}=\vec{0}-\vec{v}$. By substituting equation (3) into equation (1) we get,

(4) $\displaystyle \vec{0}+\vec{0}-\vec{v}=\vec{0}$, and thus

(5) $\displaystyle \vec{-v}=\vec{0}$.

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