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. Hence, there is no supremum for this set S in Q.

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

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