Friday, 13 November 2015

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!

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