Explanation of Supremum (sup)

This article provides a detailed explanation of the concept of “supremum (sup)” in the context of subsets \( A\subset \mathbb{R} \) of real numbers. Specifically, it describes how the supremum is defined and its properties. By understanding the intuitive meaning, mathematical formalization, and examples, we can better grasp the significance of this important concept.

What Does It Mean for a Set to Be Bounded Above?

A set \( A\subset \mathbb{R} \) is said to be bounded above if the following condition holds:
\[ \exists M \in \mathbb{R},\ \forall x \in A,\ \ \text{s.t.}\ \ x \leq M.\]

Interpretation

A set is bounded above if there exists a real number \( M \) (called an upper bound) such that every element \( x\in A \) is less than or equal to \( M \).
For example, for the set \( A=\{ 1,2,3 \} \), numbers like \( M=4,5,10 \), etc., are all upper bounds.

The Set of Upper Bounds and Supremum

For a set \( A \) that is bounded above, the set of upper bounds for \( A \) is defined as follows:
\[ \{ M^\prime \in \mathbb{R}\ ;\ \forall x \in A,\ x \leq M^\prime \}. \] Among all these upper bounds, the smallest upper bound (the minimum element of this set) is called the supremum (sup) of \( A \), denoted by \(\text{sup}\ A\).

Characteristics of the Supremum

\(\text{sup}\ A \) is an upper bound for all elements of \( A \).
\(\text{sup}\ A \) is the smallest among all upper bounds.

For example:

If \( A=\{ 1,2,3\ \} \), the supremum is \( \text{sup}\ A=3 \) (in this case, the supremum is an element of \(A\) ).
If \( A=(0,1) \), the supremum is \( \text{sup}\ A=1 \) (in this case, the supremum is not an element of \( A \)).

Two Key Properties of Supremum

The supremum \( \text{sup}\ A \) satisfies the following two conditions:

(i). All elements of \( A \) are less than or equal to the supremum
\[ \forall x \in A,\ x\leq \text{sup}\ A. \] The supremum \( \text{sup}\ A \) bounds all elements of \( A \) from above.

(ii). Any positive value \( \varepsilon > 0 \), no matter how small, allows for an element of \( A \) arbitrarily close to the supremum \[ \forall \varepsilon >0,\ \exists x \in A\ \ \text{s.t.}\ \ \text{sup}\ A-\varepsilon <x.\] There is always an element \( x\in A \) arbitrarily close to \( \text{sup}\ A \) from below.

What Does Property (ii) Mean?

Intuitive Understanding

Even if \( \text{sup}\ A \) is not itself an element of \( A \), property (ii) guarantees that there is always an element in \( A \) close to \( \text{sup}\ A \). This ensures that the supremum is “appropriately adjacent” to \( A \).

Example

(a) Set \( A=(0,1) \)

The supremum \( \text{sup}\ A=1 \) is not in \( A \). However, for any small positive \( \varepsilon >0 \), the set \( A \) contains elements greater than \(1−\varepsilon\).
- For example, if \( \varepsilon=0.1 \), elements like \( x=0.95,\ 0.99 \) in \( A \) satisfy \(1−0.1<x \leq 1\).

(b) Set \( A=\{ 1,2,3 \} \)

The supremum \( \text{sup}\ A=3 \) is an element of \( A \). Thus, for any \( \varepsilon >0 \), \(x=3\) satisfies the condition \( \text{sup}\ A−\varepsilon <x\).

This property ensures that the supremum \( \text{sup}\ A \) is always close to \( A \), even when it is not part of the set.

Visual Explanation

The properties of the supremum can be illustrated as follows:

Figure 1: Property (i)

\( \text{sup}\ A\) is an upper bound for all elements of \( A \).

Number line:
A: |----●----●----●----------| sup A
            1    2    3

Figure 2: Property (ii)

Elements of \( A \) exist arbitrarily close to \(\text{sup}\ A \).

Number line:
sup A: ------------------●
                  ↑
       sup A - ε ●    (Elements of A get arbitrarily close to sup A)

Meaning of Mathematical Symbols

The symbols used in describing the properties of the supremum are explained below:

\( \exists \): Existence quantifier (“there exists”). Corresponds to “exist” in English.
\( \forall \): Universal quantifier (“for all”). Corresponds to “for all” in English.
\( \text{s.t.} \): Condition symbol (“such that”). Abbreviated from “such that.”

Why Is Supremum Important?

The supremum \( \text{sup}\ A \) is closely related to the completeness of real numbers. The fact that the supremum exists ensures that there are no “gaps” on the real number line. This property is fundamental in analysis, number theory, and convergence of sequences.

For example, the supremum allows us to define a “maximum-like” value even for sets where a maximum does not exist. This makes the concept of the supremum one of the foundational ideas in mathematics.

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