Least Upper Bound
Definition: Let
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
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.