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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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    I've edited your question to include a link relevant to this construction and to get the subscripts. You should check your post - I am not sure that you wanted to write [(xi)]' there and in the end of the post probably one of $-x_i$ is supposed to be $-x_i'$.2011-11-08
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    For you first question: Just notice that if $z_n=-x_n$ then $|z_n-z_m|=|x_n-x_m|$ for each $n$, $m$. Then you can see directly from the definition that the sequence $(z_n)=(-x_n)$ is Cauchy.\\ Your second question is very similar: If you know that $\lim_{n\to\infty} |x_n-y_n|=0$ what can you say about sequences $(-x_n)$ and $(-y_n)$?2011-11-08
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    I would tag this as analysis more than number theory...2011-11-08

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