2.4. The Real Number System
Consider the set S of all rational numbers strictly between 0 and 1. Then
this set has many upper bounds, but only one least upper bound. That supremum
does not have to be part of the original set.
An upper bound for the set S
is any number that is greater than or
equal to any number in the set S
. Five different upper bounds
are, for example:
- 1, 10, 100, 42, and e (Euler's number)
Note that an upper bound is therefore not unique. All that is required is
to find number bigger than all other numbers in the set S
To find the least upper bound for S we need to find a number x
- x is an upper bound for S
- there is no other upper bound lower than x
Clearly, this upper bound is 1. Note that the supremum, or least upper bound,
is unique, but it is not part of the original set S.
If we include 0 and 1 in the original set S, then the least upper bound is
again 1. This time the unique supremum is part of the set S, which
may or may not happen in general.