Greatest And Least Integer Principle

The greatest and least integer principles are rather straightforward and are used a few times in the Number Theory section of Mathonline.

# Least Integer Principle

**Definition:** A nonempty set of integers that is bounded below must contain a smallest element.

# Greatest Integer Principle

** Definition:** A nonempty set of integers that is bounded above must contain a largest element.

## Example 1

Let's look at the set of positive integers, Z^{+} = {1, 2, 3, …}. Z^{+} is bounded below by 1, and hence, by the least integer principle, there must be a smallest element in the set, namely it is 1.