Completeness of $\mathbb{R}$ In the proof below I don't understand how did Terence Tao construct $L$ integer $\frac{L}{n} &lt; x_0$?
Given a nonempty set $E$ of reals, an element $x_0$ of $E$, and a positive integer $n$, OP wants to know how to construct an integer $L$ such that $L/n&lt;x_0$. In the comments, OP insists on $L$ being nonnegative. 

But this is clearly impossible, if $x_0\le0$ (and there is nothing in the problem statement forbidding such an $x_0$). For if $x_0\le0$, then, since $n\ge1$, we have $nx_0\le0$, so if $L/n&lt;x_0$ then $L&lt;nx_0\le0$, contradicting the condition that $L$ be nonnegative.
For any $\epsilon &gt; 0$ and $x \in \mathbb{R},$ there is $r$ such that $r\epsilon &gt; x$ is what you state your version of the Archimedean property is. Let's see how this implies the variant Tao uses, which is that there is an integer $K$ such that $K/n \geq M.$

By your Archimedean property, there is a real number $L \in \mathbb{R}$ such that $L \cdot (1/n) \geq M.$ The integers are unbounded in $\mathbb{R}$ (proving this will require you to be more precise on how you define $\mathbb{R}$), and so there is some $K \in \mathbb{Z}$ such that $K &gt; L.$ Hence, there is an integer $K$ with the desired property.