real analysis - Additive Inverse of Cauchy Sequence

Suppose that $x = [(x_i)]$ is a real number (equivalence class of Cauchy sequences of rational numbers).
a) How would you define the additive inverse of $x$?
b) Prove that your answer to part a is well-defined.

I know that when adding the opposite to $x$, the sum should equal zero. Is it possible to have a negative Cauchy sequence? Would the additive inverse of $x$ be $-x$?

As for proving the answer is well-defined, I believe I need to show:
If $[(x_i)]\sim[(x_i)]'$, then $[(-x_i)]\sim[(-x_i)]'$ (The i's are supposed to be sub i's.)

Suggestions would be appreciated!