The least upper bound property and the irrationals.

  • Thread starter RediJedeye
  • Start date
  • #1

Main Question or Discussion Point

Hi

Does anybody know if the irrational numbers have the least upper bound property?
 

Answers and Replies

  • #2
795
7
Hi

Does anybody know if the irrational numbers have the least upper bound property?
The set of irrationals less than zero is nonempty (it contains -pi, for example) and is bounded above (by pi, for example) yet has no least upper bound. So the irrationals do not satisfy the LUB property.

Of course zero is a LUB for that set in the reals, but 0 is not irrational. That's the beauty of the LUB concept. It encapsulates the intuition of there being "no holes" in a given set.
 
  • #3
Cool that makes perfect sense thanks for the help.
 
  • #4
Bacle2
Science Advisor
1,089
10
To generalize SteveL's answer, for any rational q,

consider the intervals (-oo,q) . Notice that the LUB of a subset of real numbers is

a limit point of that set S . So if a subset S of R does not contain all its limit points you

can constructuct a subset of S that does not contain its LUB-- so that closed subsets

contain their LUB's. Think of the relation with completeness of a set...
 

Related Threads for: The least upper bound property and the irrationals.

  • Last Post
Replies
3
Views
1K
Replies
3
Views
2K
  • Last Post
Replies
12
Views
2K
  • Last Post
Replies
1
Views
2K
Replies
1
Views
2K
Top