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### Greatest Lower Bound

Let be a nonempty set of real numbers that has a lower bound. A number is the called the greatest lower bound (or the infimum, denoted ) for iff it satisfies the following properties: 1. for all . 2. For all real numbers , if is a lower bound for , then .

### Least Upper Bound

Let be a nonempty set of real numbers that has an upper bound. Then a number is called the least upper bound (or the supremum, denoted ) for iff it satisfies the following properties: 1. for all . 2. For all real numbers , if is an upper bound for , then .

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