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Let S be a non-empty subset of R. The real number b is said to be an upper bound for S if b is greater than or equal to every element of S. If S has an upper bound, then S is bounded above. Moreover, b is a supremum (or least upper bound) of S if b is an upper bound for S and no upper bound for S is less than b; this is written b=sup S. For example, if S={0.9, 0.99, 0.999,…} then sup S=1. Similarly, the real number c is a lower bound for S if c is less than or equal to every element of S. If S has a lower bound, then S is bounded below. Moreover, c is an infimum (or greatest lower bound) of S if c is a lower bound for S and no lower bound for S is greater than c; this is written c=inf S. A set is bounded if it is bounded above and below.

It is a non-elementary result about the real numbers that any non-empty set that is bounded above has a supremum, and any non-empty set that is bounded below has an infimum.

Subjects: Mathematics.


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