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I'm trying to proof the next proposition but I have stuck in one point of the proof. I would really appreciate some help, thanks in advance.

Definition: We shall say that a Cauchy sequences is bounded away from zero, if there is a positive rational number $c$ such that $|a_n|\ge c$ for every n.

We say that $x$ positive if there is a Cauchy sequence which is positively bounded away from zero and is the formal limit of the Cauchy sequences is $x$, i.e., if $c$ is a rational number such that $c>0$, then $a_n \ge c$ for each n. Similarly $x$ is said to be negative if there is a Cauchy sequence which is negatively bounded away from zero, .i.e., $c$ is a rational number such that $c>0$, then $a_n \le -c$ for each n.

Proposition: For every real number $x$ exactly one of the following three statements is true: (a) $x$ is zero, (b) $x$ is positive or (c) $x$ is negative.

Proof: First we need to show that at most one of (a), (b) and (c) hold.

Suppose that $x$ is zero. Then $x$ can be written as a Cauchy sequences which is equivalent to the zero sequence, i.e., $\langle a_n \rangle \sim \langle 0\rangle$ then $\langle a_n \rangle $ is not bounded away from zero, and so is neither positively or negatively bounded away from zero.

Now suppose that $x$ is positive by definition there is a Cauchy sequence such that is positively bounded away from zero and its formal limit is $x$. Let $\langle \, a_n \, \rangle$ be such sequence. So, $a_n \ge c_P$ for each n where $c_P$ is a positive rational number.

By the sake of the contradiction we may assume that also $x$ is negative in other words that there exists a Cauchy sequence such that is negatively bounded away from zero and its formal limit is $x$. So $\,b_n \le -c_N$ for each n where $c_N$ is a positive rational number. Let $\,c = \text{min} (c_P ,c_N)$. Since the formal limit of both sequences is $x$ then $\langle a_n \rangle$ is equivalent to $\langle b_n \rangle$, i.e., for every $\varepsilon > 0$ there exists a $N_{\varepsilon}$ such that $|a_n-b_n| \le \varepsilon$ for each $n \ge N_{\varepsilon}$. We choose that $\varepsilon = \frac{c}{2}$. But $|a_n-b_n|$ is always positive and so $|a_n-b_n|= a_n-b_n > 2c$. Then, both sequences are not equivalent which contradicts that the formal limit of each one is $x$.

To conclude the proof we need to show that at least one of (a), (b) and (c) always hold.

Either $x$ is zero or a nonzero real number. Suppose that $x$ is a nonzero real number. Let $\langle b_n\rangle$ be the Cauchy sequence for which $x$ is the formal limit. Since $x$ is a nonzero real, $\langle b_n\rangle$ cannot be equivalent to $\langle 0\rangle$, so is not eventually $\varepsilon- \text{close}$ to the zero sequence for each $\varepsilon$. At least exists one $\varepsilon$ such that $|b_n -0| > \varepsilon$. Let us fix this $\varepsilon$.

Moreover, it is a Cauchy sequence. So we have that there is an $N$ such that $|b_k -b_j| \le \frac{\varepsilon}{2}$ for every $k,j \ge N$. On the other hand we cannot have that $|b_k|\le \varepsilon$ if so, this would imply that is eventually equivalent to the zero sequences which contradicts our assumption. Thus, there must be some $|b_k|> \varepsilon$. It follows that $|b_j| \ge \frac{\varepsilon}{2}$ for every $j \ge N$.

Here is where I'm stuck I show that is bounded away from zero but I'm not completely sure of how to show that either is positively or negatively for every $n \ge N$. Could somebody give me a hint of how to do it, please?

Thanks

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You actually need to say a little more before you choose $\epsilon$. You know that $\langle b_n\rangle\not\sim\langle 0\rangle$, so you know that there is an $\epsilon>0$ such that $|b_n|\ge\epsilon$ for infinitely many $n\in\Bbb N$; you don’t immediately know that $|b_n|>\epsilon$ for all sufficiently large $n$. To get that you must use the fact that $\langle b_n\rangle$ is Cauchy. Let $N_{\epsilon/2}$ be as in your argument, and fix $N\ge N_{\epsilon/2}$ such that $|b_N|\ge\epsilon$; then $|b_k-b_N|<\frac{\epsilon}2$ for $k\ge N$, so $|b_k|>\frac{\epsilon}2$ for $k\ge N$.

Now suppose that $k,\ell\ge N$, $b_k>0$, and $b_\ell<0$; what two contradictory conclusions can you draw about $|b_k-b_\ell|$?

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  • $\begingroup$ mmhh, since $b_k-b_\ell > 0$ clearly $|b_k-b_\ell| = b_k-b_\ell = |b_k| + |b_\ell| > \varepsilon $ which contradicts the fact that $|b_k-b_\ell|< \frac{\varepsilon}{2} $ is that right? $\endgroup$ – Jose Antonio Sep 13 '13 at 20:02
  • $\begingroup$ @Jose: Yep, you’ve got it. $\endgroup$ – Brian M. Scott Sep 13 '13 at 20:04
  • $\begingroup$ So, for each $k,\ell\ge N$, $b_k, b_\ell>0$ or $b_k, b_\ell<0$ if were not we have the above contradiction.Is that a correct conclusion? Brian as always thanks :) $\endgroup$ – Jose Antonio Sep 13 '13 at 20:09
  • $\begingroup$ @Jose: Yes: once you get out as far as $N$, all terms must be on the same side of $0$. You’re welcome! $\endgroup$ – Brian M. Scott Sep 13 '13 at 20:13
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Copying your derivation (this is part of proof of Lemma 5.3.14 in Tao's Analysis I book):

Let $ε>0$ be such that $\langle b_n \rangle$ is not eventually ε-close to zero. By $\langle b_n \rangle$ being Cauchy, there is an $N$ such that $|b_k−b_j|≤ε/2$ for every $k,j≥N$. On the other hand there exists $n_0 \geq N$ for which $|b_{n_0}| > ε$ (otherwise $b_n$ would be eventually ε-close to 0, a contradiction). So $|b_{n_0}−b_j|≤ε/2$ for all $j \geq N$.

Picking up where you left off:

Since $b_{n_0}$ is a rational number, and $|b_{n_0}| > ε >0$, we can only have exactly one of either $b_{n_0} > ε > 0$, or $b_{n_0} < -ε < 0$ (by trichotomy of rationals). Now since $\forall j \geq N$, $|b_{n_0}−b_j|≤ε/2$ (i.e., $b_j$ is only $ε/2$ away from $b_{n_0}$), it's not hard to see that if $b_{n_0} > ε > 0$, then $\forall j \geq N, b_j \geq ε/2 > 0$; similarly if $b_{n_0} < -ε < 0$, then $\forall j \geq N, b_j \leq -ε/2 < 0$. This shows that the non-zero sequence $b_n$ is eventually either positively bounded away from zero, or negatively bounded away from zero (exactly one of which is true). To finish the proof, in the first case, define $\langle a_n \rangle$ by setting $a_n:=ε/2$ for $n<N$, and $a_n:=b_n$ for $n\geq N$; in the second case, define $\langle a_n \rangle$ by setting $a_n:=-ε/2$ for $n<N$, and $a_n:=b_n$ for $n\geq N$; it's then easy to show that $\langle a_n \rangle=\langle b_n \rangle$, and $a_n$ is positively (negatively) bounded away from zero, and therefore $x$ is positive (negative).

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