Bounds, Suprema, And Infima In Real Number Sets

Let's dive into the fascinating world of real numbers and explore the concepts of upper bounds, lower bounds, suprema, and infima within a specific set. Today, we're examining the set $A = \{x \in \mathbb{R} \mid x^3 \le 3\}$. This set includes all real numbers whose cube is less than or equal to 3. Understanding these properties helps us categorize and analyze subsets of the real numbers, which is fundamental in various areas of mathematics, from calculus to abstract algebra. We'll unpack what each of these terms means and then apply them to our specific set $A$.

Understanding Upper Bounds and Lower Bounds

Before we get to our specific set, let's clarify what upper bounds and lower bounds actually are. For any given set of real numbers, say $S$, an upper bound is a number $u$ such that for every element $s$ in $S$, $s \le u$. Think of it as a ceiling – a value that the numbers in the set can never exceed. Conversely, a lower bound is a number $l$ such that for every element $s$ in $S$, $l \le s$. This is like a floor – a value that the numbers in the set can never go below. It's important to note that a set can have many upper bounds and many lower bounds, or it might have none at all. For example, the set of all real numbers $\mathbb{R}$ has no upper bound and no lower bound. However, the interval $[0, 1]$ has an upper bound of 1, 2, 100, and infinitely many others. Similarly, it has a lower bound of 0, -1, -50, and so on. The existence of bounds is a crucial characteristic of a set's structure within the real number line.

Let's consider our set $A = \{x \in \mathbb{R} \mid x^3 \le 3\}$. We are looking for numbers that are either greater than or equal to all elements in $A$, or less than or equal to all elements in $A$. To find these, we first need to get a better sense of what numbers are actually in set $A$. The condition $x^3 \le 3$ tells us that $x$ must be less than or equal to the cube root of 3. That is, $x \le \sqrt[3]{3}$. Since the cube root function is monotonically increasing, if $x^3 \le 3$, then $x \le \sqrt[3]{3}$. This means that the set $A$ is precisely the interval $(-\infty, \sqrt[3]{3}]$. Now, let's think about upper bounds for this interval. Any real number greater than or equal to $\sqrt[3]{3}$ will be an upper bound. For instance, $\sqrt[3]{3}$ itself is an upper bound because every element $x$ in $A$ satisfies $x \le \sqrt[3]{3}$. Numbers like 2, 5, or even 1,000,000 are also upper bounds for $A$. The key is that they must be greater than or equal to all values in $A$.

On the flip side, let's consider lower bounds. Since the interval extends infinitely in the negative direction $(-\infty, \sqrt[3]{3}]$, there is no real number that is less than or equal to every element in $A$. If we pick any real number, say $-M$, then we can always find an element in $A$ that is even smaller. For example, $-M-1$ is in $A$ if $(-M-1)^3 \le 3$, which is true for any large $M$. Therefore, the set $A$ has no lower bound. This is a critical observation: a set does not necessarily have to possess both upper and lower bounds. Some sets are unbounded below, some are unbounded above, and some are unbounded in both directions. Understanding this distinction is vital for applying theorems and properties related to ordered sets.

The Significance of Supremum and Infimum

While upper and lower bounds give us a general idea of a set's range, the supremum (also known as the least upper bound or LUB) and the infimum (also known as the greatest lower bound or GLB) provide more precise information. The supremum of a set $S$, denoted as $\sup(S)$, is the smallest of all the upper bounds of $S$. If a set has an upper bound, it is guaranteed to have a supremum. The supremum may or may not be an element of the set itself. Similarly, the infimum of a set $S$, denoted as $\inf(S)$, is the largest of all the lower bounds of $S$. If a set has a lower bound, it is guaranteed to have an infimum. Again, the infimum may or may not be an element of the set.

These concepts are cornerstones of real analysis because they are tied to the completeness property of the real numbers. The completeness axiom states that every non-empty subset of real numbers that is bounded above has a least upper bound (supremum) in $\mathbb{R}$. This property distinguishes the real numbers from the rational numbers, for instance. When we talk about the supremum, we are essentially pinpointing the exact