Let $I$ be a non empty interval of rational numbers with rational endpoints, i.e, I can be, open, closed, semi-open, for example:, let $a,b\in\mathbb {Q}$, $a\leq b$ and $[a,b]_{\mathbb{Q}}:=\{x\in\mathbb{Q},a\leq x\leq b\}$, the same with $<a,b>_{\mathbb{Q}}$, $<a,b]_{\mathbb{Q}}$ and $[a,b>_{\mathbb{Q}}$.
(1) Show that I has rational extremes, iff, it has supremum and infimum.
(2) Show that there is a bounded interval of rational numbers such that it is not of the 4 formats above mentioned.
- $(\Longrightarrow)$ each of the endpoints is the infimum and supremum respectively. $(\Longleftarrow)$ I have a problem here because I can not assure that the supremum and infimum are rational numbers.
Edited 2) I do not know how to begin. Let $I$ a bounded interval of rational numbers, so there is $b=\sup (I)$ and $a=\inf(I)$, then
$a\leq x\leq b$, for all $x\in I$, then $I\subset [a,b]$, is that enough? What else can I do?