Related Reading: [[Least Upper Bound Property V.S. Completeness Defined by the Convergence of Cauchy Sequences]]. The Least Upper Bound Property (also called the [[Completeness]] Property or Supremum Property) is a fundamental characteristic of the real numbers, $\mathbb{R}$. It states that: Every non-empty subset of $\mathbb{R}$ that is bounded above has a least upper bound in $\mathbb{R}$. Let's break down each part of this definition to understand what it means. ## Key Terms ### 1. Bounded Above - A set $S \subset \mathbb{R}$ is said to be bounded above if there exists a real number $U$ (an upper bound) such that for every element $x \in S$, $ x \leq U. $ - For example, the set $S = \{ x \in \mathbb{R} : 0 \leq x < 1 \}$ is bounded above by 1, since every element $x$ in $S$ is less than or equal to 1. ### 2. Upper Bound - An upper bound of a set $S \subset \mathbb{R}$ is any real number $U$ such that $x \leq U$ for all $x \in S$. ### 3. Least Upper Bound (Supremum) - The least upper bound (or supremum) of a set $S$ is the smallest real number $s$ that is an upper bound for $S$. - Formally, $s = \sup S$ if: 1. $s$ is an upper bound for $S$, meaning $x \leq s$ for all $x \in S$. 2. If $U$ is any upper bound of $S$, then $s \leq U$. - For example, in the set $S = \{ x \in \mathbb{R} : 0 \leq x < 1 \}$, the least upper bound is 1 even though 1 is not in $S$. We say that $\sup S = 1$. ## The Least Upper Bound Property of $\mathbb{R}$ The real numbers $\mathbb{R}$ are unique among number systems (like the rationals $\mathbb{Q}$) because they satisfy the Least Upper Bound Property: - If you have a non-empty subset $S \subset \mathbb{R}$ that is bounded above, there exists a smallest upper bound $s$ in $\mathbb{R}$ such that $s \geq x$ for all $x \in S$. - This property is what makes $\mathbb{R}$ complete. Not every number system, like $\mathbb{Q}$ (the rational numbers), has this property. ## Why the Least Upper Bound Property is Important The Least Upper Bound Property is fundamental to many results in real analysis and calculus, including: 1. **Existence of Limits:** It ensures that certain limits exist in $\mathbb{R}$, which is necessary for defining convergence rigorously. 2. **Cauchy Sequences:** It’s the property that allows us to say every Cauchy sequence converges to a real number. 3. **Development of Integration and Differentiation:** Many theorems in calculus (such as the Intermediate Value Theorem and Extreme Value Theorem) rely on the completeness of $\mathbb{R}$, which in turn depends on the Least Upper Bound Property. ## Example to Illustrate the Property Consider the set $S = \{ x \in \mathbb{R} : x^2 < 2 \}$, which represents all real numbers whose square is less than 2. - This set $S$ is bounded above. For example, 2 is an upper bound because any $x$ with $x^2 < 2$ must be less than 2. - However, 2 is not the smallest upper bound. The least upper bound of $S$ is $\sqrt{2}$, because: 1. For all $x \in S$, $x \leq \sqrt{2}$. 2. Any number smaller than $\sqrt{2}$ (say, 1.4) is not an upper bound of $S$ since there are elements in $S$ (like 1.41) that are greater than 1.4. So, by the Least Upper Bound Property, we know that $\sup S = \sqrt{2}$ exists in $\mathbb{R}$. ## Summary The Least Upper Bound Property of $\mathbb{R}$ is what guarantees that every non-empty, bounded-above subset of real numbers has a smallest upper bound. This property is essential for the completeness of $\mathbb{R}$, which in turn is a foundation for much of real analysis and calculus.