# Nonempty Subset of Q

• May 5th 2011, 08:51 AM
jstarks44444
Nonempty Subset of Q
Find a nonempty subset of Q (rational numbers) that is bounded above but has no least upper bound in Q. Justify your claim. Thanks for the help!
• May 5th 2011, 08:56 AM
TheEmptySet
Quote:

Originally Posted by jstarks44444
Find a nonempty subset of Q (rational numbers) that is bounded above but has no least upper bound in Q. Justify your claim. Thanks for the help!

$\{x| (x \in \mathbb{Q}) \cap (0 \le x^2 < 2) \}$
• May 5th 2011, 08:57 AM
Plato
Quote:

Originally Posted by jstarks44444
Find a nonempty subset of Q (rational numbers) that is bounded above but has no least upper bound in Q. Justify your claim. Thanks for the help!

This is the quintessential example: $\{q\in\mathbb{Q}:q^2<2\}$
Now the details of the proof are yours to do.
• May 6th 2011, 07:12 PM
Deveno
in general, any set Q ∩ (a,b) (where (a,b) is an open interval of the reals) where b is irrational will do. why is this so?