# Proposition 5.4.9. Analysis I - Terence Tao.

Proposition 5.4.9 (The non-negative reals are closed). Let $$a_1, a_2, a_3, \ldots$$ be a Cauchy sequence of non-negative rational numbers. Then $$\text{LIM}_{n \to \infty}a_n$$ is a non-negative real number.

Tao's Proof.

We argue by contradiction, and suppose that the real number $$x:= \text{LIM}_{n \to \infty}a_n$$ is a negative number. Then by definition of negative real number, we have $$x:= \text{LIM}_{n \to \infty}b_n$$ for some sequence $$b_n$$ which is negatively bounded away from zero, i.e, there is a negative rational $$-c < 0$$ such that $$b_n \leq -c$$ for all $$n \geq 1$$. On the other hand, we have $$a_n \geq 0$$ for all $$n \geq 1$$, by hypothesis. Thus the numbers $$a_n$$ and $$b_n$$ are never $$c/2$$-close, since $$c/2 < c$$. Thus the sequences $$(a_n)_{n=1}^{\infty}$$ and $$(b_n)_{n=1}^{\infty}$$ are not eventually $$c/2$$-close. Since $$c/2 > 0$$, this implies that $$(a_n)_{n=1}^{\infty}$$ and $$(b_n)_{n=1}^{\infty}$$ are not equivalent. But this contradicts the fact that both these sequences have $$x$$ as their formal limit.

My Question.

I was surprised to see Tao's proof was more complicated than mine. Where we differed was that I did not introduce a second sequence $$(b_n)_{n=1}^{\infty}$$. Why did he need to do this? Couldn't we just work with $$(a_n)_{n=1}^{\infty}$$? We would arrive at showing $$a_n \leq -c < 0$$ for all $$n \geq 1$$, which would contradict that every $$a_n$$ is non-negative.

In Chapter 5 Tao introduces the real numbers as formal expressions $$\operatorname{LIM}_{n \to \infty} a_n$$, where $$(a_n)$$ is a Cauchy sequence of rational numbers. Two real numbers $$\operatorname{LIM}_{n \to \infty} a_n$$ and $$\operatorname{LIM}_{n \to \infty} b_n$$ are said to be equal iff $$(a_n)$$ and $$(b_n)$$ are equivalent Cauchy sequences. This means actually that $$\operatorname{LIM}_{n \to \infty} a_n$$ denotes the equivalence class of $$(a_n)$$ with respect to the equivalence relation "equivalent Cauchy sequences". See Confusion about Tao's construction of reals.
Thus $$\operatorname{LIM}_{n \to \infty} a_n$$ does not denote the "usual limit" of the sequence $$(a_n)$$. This concept is introduced only later in Chapter 6. The limit of a sequence $$(x_n)$$ of real numbers is written as $$\lim_{n \to \infty} x_n$$.
This explains why his proof uses the second sequence $$(b_n)$$. It is a different representative of the real number $$x = \operatorname{LIM}_{n \to \infty} a_n$$ and he shows that the assumption on $$(b_n)$$ leads to a contradiction.
Your proof comes too early for the status achieved in Chapter 5. Actually you work with $$\lim_{n \to \infty} a_n$$ and not with the formal expression $$\operatorname{LIM}_{n \to \infty} a_n$$.