I'm trying to find the supremum and infimum of this set .
$$I = ]-\pi; \pi[ \cap Q$$
The problem is that it accept a least upper bound (resp : greatest lower bound) since it's a bounded set and non empty . However there's an endless rational numbers that are bigger than all elements of this set due to the density of $Q$ in $R$ , also from the archimdean property we know that for every $\epsilon > 0$ there exist some rational number smaller than $\epsilon$ . Which mean that I can always find a smaller number then any rational I suppose to be the supremum
Can anyone explain why this has or hasn't any $sup$ or $inf$ while it's a non empty set and bounded ?