## 2.4. The Real Number System

### Examples 2.4.3(b):

Consider the set of rational numbers {1, 1.4, 1.41, 1.414, 1.4142, ...}
converging to the square root of 2. If all we knew were rational numbers,
this set would have no supremum. If we allow real numbers, there is
a unique supremem.

If we consider the universe to consist only of rational numbers, then
this set does not have a least upper bound.
- No number bigger than
is the least upper bound (although each of these numbers is an upper bound),
because if
*x*was that least upper bound, then we can find a rational number between and*x*. That rational number would then be an upper bound smaller than*x*, which is a contradiction. - No number less than
is the least upper bound, because if
*x*was that least upper bound, there is some element of the set between*x*and . But then*x*is not an upper bound, which is a contradiction.

**S**in

**Q**.

If we consider this set as a subset of the real numbers, then the least upper bound of this set is .