Q:

How to prove (-1)*u=-u for all real numbers $u$?

How can we show that additive inverse of a real number equals the number multiplied by -1, i.e. how can we show that $(-1)*u = -u$ for all  real numbers $u$?

A:

$-u$ is the unique additive inverse of $u$. By distributivity we have 
$$(-1)u+u=(-1)u+(1)u=(-1+1)u=0u=0$$
and hence $(-1)u=-u.$ To prove $0u=0$, observe $0u=(0+0)u=0u+0u$ and the additive identity element is unique.

