## Quick Reference

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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