I am going through Analysis I by Terence Tao and am currently stuck at proving this proposition, which is as follows:
Proposition 5.4.4 (Basic properties of positive reals). For every real number $x$, exactly one of the following three statements is true: (a) $x$ is zero; (b) $x$ is positive; (c) $x$ is negative.
How, I have proceeded so far:
If suppose $x$ is $0$,then the claim easily follows, therefore suppose it isn't then it must be bounded away from zero. If $x=LIM_{n\to \infty}a_n$, this must then mean that $(a_n)_{n=1}^{\infty}$ is a Cauchy sequence of rationals and that there exists a positive rational $c$ such that $|a_n|\gt c$ for all natural $n$. But since $(a_n)_{n=1}^{\infty}$ is a Cauchy sequence then there must be a natural $N$ such that $|a_j-a_k|\leq \frac{c}{2}$, for all $j,k\geq N$.
As I can visualise this then means the sequence will become positive or negative thereafter. But how can I continue after this? What arguments should I put forth?
As for the book that I am following the real numbers are defined using formal limits (hence $LIM$ and not $\lim$). The author does state that formal limits will eventually be superceded by the actual limits but I haven't reached upto that point so I'm avoiding to use the notation for the same. For now consider a real number to be any object of form $LIM_{n\to \infty}a_n$ where $(a_n)_{n=1}^{\infty}$ is a Cauchy sequence of rationals and two reals are equal if and only if their respective sequences are equivalent.
