Find and Prove the inf and sup:
{x is a rational number: x^2 < 2 }
If you were taking x in the real numbers, then your supremum would be and your infimum would be .
However, these are both irrational numbers. There does not exist a supremum (least upper bound) in the rational numbers. Say it did exist, and call it a. Then . Define . Then b is also an upper bound of your set, and , which contradicts our assumption that a was our LEAST upper bound.
A similar argument works for the infimum.