Supremum and Infimum Explained with Graphs: Definition, Properties, and Applications

Supremum and Infimum — Complete Guide With Definitions, Properties & Applications

In mathematics, especially in real analysis, the concepts of supremum and infimum are fundamental tools used to describe bounds of sets, functions, and sequences. Mastering these concepts helps in understanding the behavior of functions, analyzing limits, studying series convergence, and solving optimization problems.

What Are Supremum and Infimum?

Supremum (sup)

The supremum of a set or function is the smallest number that is greater than or equal to all elements in that set or the range of the function. It is also referred to as the least upper bound.

• If a set has a maximum element, the supremum is the maximum.

• Even if the maximum does not exist in the set, the supremum still exists if the set is bounded above.

Example:For the set S={x∈R:x<5}S = \{x \in \mathbb{R} : x < 5\}S={x∈R:x<5}, the supremum is 5, even though 5 itself is not in the set.

Infimum (inf)

The infimum of a set or function is the largest number that is less than or equal to all elements in the set or function range. It is also called the greatest lower bound.

• If a set has a minimum element, the infimum is the minimum.

• If no minimum exists, the infimum still exists if the set is bounded below.

Example:For the set T={x∈R:x>2}T = \{x \in \mathbb{R} : x > 2\}T={x∈R:x>2}, the infimum is 2, even though 2 is not included in the set.

Why Supremum and Infimum Matter

Supremum and infimum are crucial in situations where a maximum or minimum may not exist within a set. They provide precise bounds for:

• Sequences: Understanding convergence and limits.

• Functions: Identifying least upper or greatest lower bounds in optimization problems.

• Sets: Analyzing boundedness and completeness in real numbers.

These concepts also underpin fundamental theorems in analysis, such as the least upper bound property of real numbers.

This image visually explains the concepts of supremum (sup) and infimum (inf) for a real-valued function ff:

• The orange curve, labeled as ff, represents the graph of a function across a certain interval (its domain, marked along the horizontal axis).

• The domain of f (bottom horizontal bracket, labeled in red) indicates the range of input values over which the function is defined.

• The range of f (left vertical bracket, labeled in yellow) represents all the possible output values that ff can take for its domain.

• The green point labeled supₐ f marks the supremum (least upper bound) of the function over its domain. This is the smallest value that is greater than or equal to every value the function attains.

• The blue point labeled infₐ f marks the infimum (greatest lower bound) of ff on its domain. This is the largest value that is less than or equal to every value the function takes.

• Note that in some cases, the supremum and infimum might be values that the function does not actually attain but that serve as tightest possible upper and lower bounds for the range.

• This kind of visualization highlights the difference between maximum/minimum (which must be reached) and supremum/infimum (which may simply bound the outputs).

Overall, the image shows how, by examining extremes in both the input (domain) and output (range), mathematicians formally define and locate the bounds of a function using supremum and infimum.

Visual Explanation

Consider a function f(x)f(x)f(x) plotted on a graph:

• The domain (x-axis) represents all input values.

• The range (y-axis) represents all output values of fff.

• Sup fff: Marks the highest value that the function’s outputs approach or attain.

• Inf fff: Marks the lowest value that the outputs approach or attain.

Even if the function never exactly reaches these values, they define the bounds within which the function operates.

Example Visual Scenario:

• A function increases toward a horizontal asymptote at y=10y = 10y=10 without ever touching it. Here, the supremum of the function is 10.

• If the function decreases toward a horizontal line at y=−2y = -2y=−2, the infimum is -2.
  

Summary

• Supremum (sup): Least upper bound; the smallest number ≥ all set elements.

• Infimum (inf): Greatest lower bound; the largest number ≤ all set elements.

• These concepts are vital in real analysis, limits, sequences, series, and optimization problems.

• Supremum and infimum allow us to understand bounds even when maxima or minima don’t exist in a set or function.

Understanding these terms helps in visualizing bounded behavior, analyzing function outputs, and solving optimization problems in mathematics and applied fields. Supremum and infimum are not just abstract mathematical concepts—they have specific properties, real examples, and wide-ranging applications in mathematics and applied sciences. Understanding them thoroughly helps in analyzing functions, sets, and sequences, especially when maxima or minima do not exist within a set.

Properties of Supremum and Infimum

1. Existence in Bounded Sets

   • Any non-empty set that is bounded above has a supremum.

   • Any non-empty set that is bounded below has an infimum.This ensures that we can always identify least upper and greatest lower bounds for sets that don’t extend infinitely.

2. Connection to Maximum and Minimum

   • If the supremum of a set is actually a member of the set, it is also the maximum.

   • Similarly, if the infimum belongs to the set, it is the minimum.

3. Completeness of Real Numbers

   • Supremum and infimum are foundational in defining the completeness property of real numbers. This property states that every non-empty set of real numbers that is bounded above has a supremum, and every non-empty set bounded below has an infimum.

   • This completeness is crucial for rigorous analysis, limits, and convergence in calculus.

Examples

1. Function Example:

f(x)=1x,x∈(0,∞)f(x) = \frac{1}{x}, \quad x \in (0, \infty)f(x)=x1​,x∈(0,∞)

• As xxx approaches 0, f(x)f(x)f(x) grows without bound, so sup⁡f=∞\sup f = \inftysupf=∞.

• As xxx increases toward infinity, f(x)f(x)f(x) approaches 0, so inf⁡f=0\inf f = 0inff=0.

2. Finite Set Example:

A={2,4,6,8}A = \{2, 4, 6, 8\}A={2,4,6,8}

• Supremum (sup⁡A\sup AsupA) = 8

• Infimum (inf⁡A\inf AinfA) = 2

3. Infinite Set Example:

B={1/n:n∈N}B = \{1/n : n \in \mathbb{N}\}B={1/n:n∈N}

• Supremum (sup⁡B\sup BsupB) = 1 (largest value in the set)

• Infimum (inf⁡B\inf BinfB) = 0 (approaches zero as n→∞n \to \inftyn→∞ but never reaches it)

These examples illustrate how supremum and infimum generalize maximum and minimum, even in cases where the extreme values are not part of the set or function.

Applications

1. Calculus and Analysis

   • Supremum and infimum are used to evaluate limits, continuity, and boundedness of functions.

2. Series and Sequences

   • They help in determining convergence and the radius of convergence of power series.

3. Mathematical Optimization

   • In optimization problems, supremum and infimum identify solution boundaries, especially for functions without explicit maxima or minima.

4. Advanced Mathematical Theory

   • These concepts are foundational in real analysis, providing a rigorous framework for the completeness of real numbers and handling infinite or unbounded scenarios.

Conclusion

Mastering supremum and infimum is essential for advanced mathematical studies and practical applications in science, engineering, and economics. They extend the concepts of maximum and minimum to more general settings, allowing mathematicians to rigorously handle sets and functions that are infinite, unbounded, or lack explicit extremes. Understanding these principles equips students and professionals with powerful tools for analyzing, predicting, and optimizing in both theoretical and applied contexts. 20 MCQs on the topic of Supremum and Infimum with answers and explanations:

1. What is the supremum of a set?

A) The smallest element in the set

B) The smallest number greater than or equal to all elements

C) The largest element in the set

D) The largest number less than all elements

Answer: B

Explanation: Supremum is the least upper bound, not necessarily in the set.

2. What is the infimum of a set?

A) The greatest element in the set

B) The largest number less than or equal to all elements

C) The smallest element in the set

D) The least number greater than all elements

Answer: B

Explanation: Infimum is the greatest lower bound, which may or may not be in the set.

3. Which of the following is true?

A) Every set has a maximum

B) Every set has a supremum

C) Supremum is always in the set

D) Infimum is always less than the smallest element

Answer: B

Explanation: Every non-empty set bounded above has a supremum.

(0,∞), what is the infimum?

A) 0

B) 1

C) Infinity

D) Undefined

Answer: A

Explanation: The function values approach 0 but never reach it, so infimum is 0.

6. Does the supremum have to be a member of the set?

A) Yes

B) No

Answer: B

Explanation: Supremum can be a limit point not in the set.

7. What does

inf

⁡

S

infS denote?

A) Least element of set

S

S

B) Greatest element of set

S

S

C) Greatest lower bound of

S

S

D) Least upper bound of

S

S

Answer: C

Explanation: Infimum is the greatest number less than or equal to all elements of

S

S.

8. What can be said about a set with no upper bound?

A) It does not have a supremum

B) It always has an infimum

C) It has a supremum

D) Both of the above are false

Answer: A

Explanation: Supremum requires an upper bound.

9. If

sup

⁡

A

=

max

⁡

A

supA=maxA, then the supremum:

A) Is outside the set

B) Is the largest element of the set

C) Does not exist

D) Is equal to the infimum

Answer: B

Explanation: If supremum is in

A

A, it equals the maximum.

10. What type of number system guarantees the existence of suprema and infima for bounded sets?

A) Rational numbers (

Q

Q)

B) Real numbers (

R

R)

C) Integer numbers

D) Complex numbers

Answer: B

Explanation: Completeness property of

R

R guarantees sup and inf.

11. The limit superior of a sequence is related to:

A) The supremum of limit points

B) The same as the sequence maximum

C) The infimum of the sequence

D) The last term in the sequence

Answer: A

Explanation: Limit superior is the supremum of all subsequential limits.

12. What is true about infimum and supremum?

A) Infimum is always smaller than supremum

B) Infimum and supremum are equal for all sets

C) Supremum can be less than some elements of set

D) Infimum can be higher than some elements of set

Answer: A

Explanation: By definition, infimum ≤ supremum.

13. The term "bounded below" means:

A) No lower bound exists

B) There exists an element that is less than or equal to all others

C) The set is finite

D) The supremum of the set is finite

Answer: B

Explanation: There is a number that bounds the set from below.

15. Which tool helps quantify the difference between two sets in terms of their supremum?

A) Distance function

B) Supremum distance

C) Radius of convergence

D) Linear transformation

Answer: B

Explanation: Supremum distance measures maximal gap between sets.

17. To prove a number

x

x is the supremum, one must show:

A)

x

x is in the set

B)

x

x is an upper bound and any smaller number is not an upper bound

C)

x

x is a lower bound

D)

x

x is the average of the set

Answer: B

Explanation: Supremum must be the smallest upper bound.

18. The concept of supremum and infimum is essential for:

A) Completing the real numbers

B) Algebra only

C) Geometry only

D) Complex numbers only

Answer: A

Explanation: These concepts underpin completeness of the real number system.

19. What is the supremum of the empty set?

A) 0

B) Does not exist

C) Infinity

D) -Infinity

Answer: B

Explanation: The empty set has no elements to bound it.

20. If a set contains its supremum, the supremum is also:

A) Infimum

B) Minimum

C) Maximum

D) None of the above

Answer: C

Explanation: Supremum in the set is the maximum element.

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