Inverses for Real Addition

Theorem

Each element $x$ of the set of real numbers $\R$ has an inverse element $-x$ under the operation of real number addition:

$\forall x \in \R: \exists -x \in \R: x + \paren {-x} = 0 = \paren {-x} + x$

Proof

We have:

\(\displaystyle \eqclass {\sequence {x_n} } {} + \paren {-\eqclass {\sequence {x_n} } {} }\)

\(=\)

\(\displaystyle \eqclass {\sequence {x_n - x_n} } {}\)

\(\displaystyle \)

\(=\)

\(\displaystyle \eqclass {\sequence {0_n} } {}\)

Similarly for $\paren {-\eqclass {\sequence {x_n} } {} } + \eqclass {\sequence {x_n} } {}$.

Thus the inverse of $x \in \struct {\R, +}$ is $-x$.

$\blacksquare$