Least Upper Bound

Definition: Let be a subset of , and be a real number. We say that is a least upper bound for iff

  • is an upper bound for
  • any other upper bound for must be larger than or equal to .

Least Upper Bound Property

Empty set does not have a least upper bound. There always exist exact one least upper bound for any non-empty set that has an upper bound (existence and uniqueness).

Least upper bound property can be used to develop the idea of limit

Supremum

Define to be the least upper bound of if has a least upper bound. If is a non-empty set with no upper bound, . If is an empty set, .

Greatest Lower Bound and Infimum

The definitions of /greatest lower bound/ and /infimum/ are similar to above.

Prove that there is a Real Number such that

  • Construct a set to be all none-negative real numbers whose square root is less than 2
  • We can apply the least upper bound property because is not empty () and is upper bounded by 2
  • We get , and we prove
  • We can prove by contradiction. For example, if , we can find a small number s.t.