I want to prove $u + v = v + u$, where $u$ and $v$ are vectors. I've seen the proof where you apply the vector rule for addition but I was wondering if this proof also works.
$$u + v = v + u $$
$$0 = (v + u) - (u + v)$$
$$0 = (v_1 - v_1 + ... + v_n - v_n) + (u_1 - u_1 + ... + u_n - u_n)$$ $$ 0 = 0$$ therefore the property is proven.