I'm reading this question (Terence Tao Exercise 5.4.8: Boundedness of Limit.) regarding the proof a property of Cauchy sequences. However, I don't understand a step in the proof,
... suppose $a = \text{LIM}_{n\rightarrow \infty}a_n >x$, there exists a rational $q$ that $a>q>x$ due to Prop 5.4.14. Since $\forall n\geq 1,a_n\leq x<q$, construct a sequence $(q−an)$, it's trivial to prove the sequence is Cauchy and it's non-negative...
I don't understand why is $(q−a_n)$ non-negative, because, $$a_n\leq x<q, \qquad n\geq 1,$$ $$\Rightarrow a_n<q ,$$ $$\Rightarrow 0<(q-a_n).$$
Hence the sequence is strictly positive, this is critical in the next step because the following theorem is valid for non-negative sequences only.
I'm also confused with the use of the "$\geq$" and "$>$" in a few proofs of real analysis, for example in page (117,118) of Terence Tao Analysis I, when he proves the (Existence of least upper bound), he performs the following step,
How can he conclude that,
$$\Big| \frac{m_n}{n} - \frac{m_{n}}{n'} \Big|\leq \frac{1}{N} \quad \text{for all } \quad n,n' \geq N\geq 1,$$
since in a previous equation he had:
$$ \frac{m_n}{n} - \frac{m_{n}}{n'} > -\frac{1}{N} $$
Can one just put the "$\geq$" instead of the "$>$" ?