MathCS.org - Real Analysis: Examples 2.4.3(a):

Examples 2.4.3(a):

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.

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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 for S 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 such that

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.

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Examples 2.4.3(a):