I am working through Tao's analysis and I am trying to prove theorem 5.5.9 However, I have some trouble with the following line:
"Let $n \geq 1$ be a positive integer. We know that $E$ (where $E \subset \mathbb{R}$) has an upper bound $M$. By the Archimedean property we can find an integer $K$ such that $K/n \geq M$."
So far he has introduced $E$ which is a subset of $\mathbb{R}$ and $M$ which is an upper bound of $E$. However, as far as I understand the content, letting $n \in \mathbb{Z}^+$ is not sufficient to use the Archimedean property. Indeed this property is stated in the very same book as:
\begin{equation}
x,\epsilon \in \mathbb{R}^+ \rightarrow \exists M\in \mathbb{Z}^+(M\epsilon > x)
\end{equation}
Nevertheless, in this case we do not know if $M \in \mathbb{R}^+$. By definition of uppper bound we only know $M \in \mathbb{R}$. Letting $n \in \mathbb{Z}^+$ allows one to say that $1/n \in \mathbb{R}^+$, but I do not see whay $M$ should be positive and why one can use the Archimedean property.
In particular, the three elements in the relation are $M,n,K$. However, by the text only one of these ($n$) is positive, while the Archimedean property, as stated requires two elements to be positive. Is he assuming something else? Is it the case, that it also works for general real numbers but the is not showing it?
Best Answer
If $M$ is not positive, applying the Archimedean property to $x=1$ and $\epsilon=\frac1n$, you get a $K$ with $$ K\epsilon>x>M $$ and particularly, $K\epsilon>M$.