The rational numbers have closure properties under addition and multiplication, meaning that the sum or product of any two rational numbers is also a rational number. However, when it comes to finding roots, the rational numbers do not exhibit closure. This can be exemplified by considering the equation x^2 = 2. It is evident that this equation does not have a rational root.
Before delving into the main objective of this lesson, let’s revisit a few concepts. Let S be a non-empty set of real numbers. If there exists a number M such that x ≤ M for every x in S, then the set S is said to be bounded above. In this case, M is referred to as an upper bound of S.
On the other hand, if there exists a number m such that x ≥ m for all x in S, then the set S is said to be bounded below. The number m is called the lower bound of S.
If there exists a number M such that the absolute value of x is less than or equal to M for all x in S, then the set S is bounded. Here, the number M is called a bound of S. A set that is not bounded is referred to as unbounded. To understand the concept of an unbounded set, we can simply negate the definition of what it means to be bounded.
For example, a set S is unbounded if, for any number M, there exists an element x in S such that the absolute value of x is greater than M.
Now, let’s move on to defining the concept of supremum. Suppose S is a set that is bounded above. The number β is called the supremum of S if β is an upper bound of S and no number less than β is an upper bound. In other words, β is the smallest upper bound.
Conversely, if S is bounded below, the number α is referred to as the infimum of S if α is a lower bound of S and no number greater than α is a lower bound. We denote this by saying that α is the infimum. It is important to note that a set can only have at most one infimum.
The proof of this statement is left as an exercise. The supremum of a set is sometimes called the least upper bound, while the infimum of a set is often referred to as the greatest lower bound.
The Completeness Axiom states that any nonempty subset of the real numbers that is bounded above has a supremum.