# Supremum principle

## Main Question or Discussion Point

According to supremum and infimum principle,
nonempty set A={x|x$$\in$$Q,x2<2} is upper bounded, so it should have a least upper bound. In fact, it dose not have least upper bound. Why?
When the principle is valid?

Last edited:

LCKurtz
Homework Helper
Gold Member
Assuming you are talking about the rationals Q, your set isn't even defined in terms of elements of Q. You should phrase it as x2 < 2. Why should it have a least upper bound? There is no theorem stating that a subset of the rationals Q which is bounded above has a least upper bound in Q. In fact, one way to develop the real numbers is to extend them by Dedekind cuts which, effectively, adds all such upper bounds and gives the reals R. Such subsets viewed as subsets of R do have least upper bounds in R.

HallsofIvy
Homework Helper
According to supremum and infimum principle,
nonempty set A={x|x$$\in$$Q,x2<$$\sqrt{2}$$} is upper bounded, so it should have a least upper bound. In fact, it dose not have least upper bound. Why?
When the principle is valid?
First, I suspect you have not written the set correctly. I believe you meant $A= \{x | x\in Q, x^2< 2\}$. As a set of real numbers, that does have a least upper bound- it is $\sqrt{2}$. Since that is not rational, if you think of that set as a subset of the rational numbers, in does not have a least upper bound (in the rational numbers). The "supremum and infimum property" does not hold for the set of rational numbers. In fact, it is a "defining property" of the real numbers.

The "supremum and infimum property" does not hold for the set of rational numbers.
what does "the set of rational numbers" mean? Is {1,2,3} a set of rational numbers?

LCKurtz
Homework Helper
Gold Member
what does "the set of rational numbers" mean? Is {1,2,3} a set of rational numbers?
You're taking a course talking about sups and infs and you don't know what the rational numbers are???

{1,2,3} is a set of three positive integers (they are also rational numbers).
{1,2,3,...} would be the set of positive integers
{...,-3,-2,-1,0,1,2...} would be the set of integers
x is a rational number if x can be expressed as the quotient of two integers.

HallsofIvy