Supremum and Infimum: Finding the Best Bounds

One of the first questions my real analysis students ask is: "What's the difference between maximum and supremum?" This confusion is natural, but understanding the distinction is crucial for rigorous analysis!

Upper and Lower Bounds First

Before we define supremum and infimum, we need to understand bounds.

Upper Bound: A number M is an upper bound of set A if x ≤ M for all x ∈ A.

Lower Bound: A number m is a lower bound of set A if m ≤ x for all x ∈ A.

Example: For A = (0, 1)

  • 1, 2, 5, 100 are all upper bounds
  • 0, -1, -5, -100 are all lower bounds

Notice: A set can have infinitely many bounds!

What is Supremum?

The supremum (or least upper bound) of set A, denoted sup A or lub A, is the SMALLEST upper bound.

Definition: sup A = M if:

  1. M is an upper bound: x ≤ M for all x ∈ A
  2. M is the smallest such bound: if M' is any upper bound, then M ≤ M'

Alternatively: For any ε > 0, there exists x ∈ A such that M − ε < x ≤ M

Think: "The tightest upper bound possible"

What is Infimum?

The infimum (or greatest lower bound) of set A, denoted inf A or glb A, is the LARGEST lower bound.

Definition: inf A = m if:

  1. m is a lower bound: m ≤ x for all x ∈ A
  2. m is the largest such bound: if m' is any lower bound, then m' ≤ m

Alternatively: For any ε > 0, there exists x ∈ A such that m ≤ x < m + ε

Think: "The tightest lower bound possible"

Understanding Through Examples

Example 1: The Open Interval (0, 1)

Does (0, 1) have a maximum? No! We can always find a larger element.

  • 0.9 ∈ (0, 1), but 0.99 > 0.9
  • 0.99 ∈ (0, 1), but 0.999 > 0.99
  • No largest element exists

But does it have a supremum? Yes!

  • sup(0, 1) = 1
  • 1 is an upper bound (all elements are less than 1)
  • No smaller number works (1 − ε is NOT an upper bound for any ε > 0)

Similarly:

  • inf(0, 1) = 0
  • 0 is a lower bound
  • No larger number works

Notice: sup(0, 1) = 1 ∉ (0, 1) and inf(0, 1) = 0 ∉ (0, 1)

Example 2: The Closed Interval [0, 1]

  • sup[0, 1] = 1 (and 1 ∈ [0, 1])
  • inf[0, 1] = 0 (and 0 ∈ [0, 1])
  • max[0, 1] = 1 (exists!)
  • min[0, 1] = 0 (exists!)

Here supremum equals maximum, and infimum equals minimum!

Example 3: The Set {1, 1/2, 1/3, 1/4, ...}

  • sup{1, 1/2, 1/3, ...} = 1 (the first element)
  • inf{1, 1/2, 1/3, ...} = 0 (limit of the sequence, NOT in the set!)

Notice: 0 is NOT in the set, but it's still the infimum!

Example 4: The Set {−1, −2, −3, −4, ...}

  • This set has no upper bound, so sup doesn't exist (or we write sup = ∞)
  • inf{−1, −2, −3, ...} doesn't exist in ℝ (or we write inf = −∞)

Example 5: The Rationals Between 0 and √2

A = {r ∈ ℚ | 0 < r < √2}

  • sup A = √2 (even though √2 ∉ ℚ!)
  • inf A = 0
  • No maximum exists in ℚ (since √2 is irrational)

This shows supremum can exist even when maximum doesn't!

Supremum vs Maximum

This is where students often get confused!

Maximum: The largest element IN the set (if it exists)

Supremum: The least upper bound (may or may not be in the set)

Key Differences:

  • Maximum must belong to the set
  • Supremum may or may not belong to the set
  • If maximum exists, then supremum = maximum
  • Supremum can exist even when maximum doesn't!

Example: A = (0, 1)

  • max A does NOT exist (no largest element in the set)
  • sup A = 1 (exists, but 1 ∉ A)

Example: A = [0, 1]

  • max A = 1 (exists and equals sup A)
  • sup A = 1

Similarly for Infimum vs Minimum:

  • Minimum must belong to the set
  • Infimum may or may not belong to the set
  • If minimum exists, then infimum = minimum

The Completeness Axiom

This is THE fundamental property of real numbers!

Completeness Axiom: Every non-empty set of real numbers that is bounded above has a supremum (in ℝ).

Similarly: Every non-empty set that is bounded below has an infimum (in ℝ).

This is what makes ℝ "complete" – no gaps!

Counterexample in ℚ: The set {r ∈ ℚ | r² < 2} is bounded above in ℚ, but has NO supremum in ℚ (since √2 ∉ ℚ). It does have sup = √2 in ℝ!

Properties of Supremum and Infimum

Property 1: If A ⊆ B and both have suprema, then sup A ≤ sup B

Property 2: For any c > 0, sup(cA) = c · sup A where cA = {cx | x ∈ A}

Property 3: sup(A + B) = sup A + sup B where A + B = {a + b | a ∈ A, b ∈ B}

Property 4: If sup A exists and a ∈ A, then a ≤ sup A

Property 5: If A is finite and non-empty, then sup A = max A and inf A = min A

How to Find Supremum and Infimum

Step 1: Determine if the set is bounded above/below

Step 2: Make an educated guess for sup/inf

Step 3: Verify two conditions:

  • Is it an upper/lower bound?
  • Is it the smallest/largest such bound?

Alternative for Step 3: Use the ε-characterization

  • For sup: Show that for any ε > 0, there exists x ∈ A with M − ε < x

Example: Find sup{1 − 1/n | n ∈ ℕ}

Guess: sup = 1

Verification:

  • Is 1 an upper bound? Yes: 1 − 1/n < 1 for all n ✓
  • Is 1 the smallest? For any ε > 0, take n > 1/ε. Then 1 − 1/n > 1 − ε ✓

Therefore sup = 1 (and 1 is NOT in the set!)

Practice Problem 1

Question: Find sup A and inf A where A = {(−1)ⁿ/n | n ∈ ℕ}

Answer: A = {−1, 1/2, −1/3, 1/4, −1/5, ...}

  • sup A = 1/2 (the largest element, which IS in A)
  • inf A = −1 (the smallest element, which IS in A)

Practice Problem 2

Question: Find sup{x ∈ ℚ | x² < 3}

Answer: sup = √3

Even though √3 ∉ ℚ, it is the supremum in ℝ!

Practice Problem 3

Question: Does the set {1/n − 1/m | n, m ∈ ℕ} have a supremum and infimum?

Answer:

  • When n = 1, m → ∞: 1 − 1/m → 1
  • When m = 1, n → ∞: 1/n − 1 → −1

So sup = 1 and inf = −1 (neither is in the set!)

Multiple Choice Quiz

Question: For which set does supremum equal maximum?

A) (0, 1)
B) (0, 1]
C) {1/n | n ∈ ℕ}
D) ℚ ∩ (0, 1)

Answer: B) (0, 1]

Explanation:

  • A) sup = 1, but no max exists
  • B) sup = max = 1 ✓
  • C) sup = max = 1 ✓ (wait, this also works!)
  • D) sup = 1, but 1 ∉ ℚ ∩ (0, 1), so no max

Actually both B and C are correct! Let me reconsider... In C, 1 ∈ {1/n | n ∈ ℕ} when n = 1, so max exists and equals 1.

Common Mistakes Students Make

Mistake 1: Confusing supremum with maximum Reality: Supremum may not be in the set!

Mistake 2: Thinking every bounded set has a maximum Reality: (0, 1) is bounded but has no maximum

Mistake 3: Assuming sup A must be greater than all elements Reality: sup A ≥ all elements (can be equal to the largest element)

Mistake 4: Forgetting that ∅ has no supremum or infimum Reality: The empty set has no bounds

Why Supremum and Infimum Matter

In Sequences: Bounded sequences have supremum and infimum, leading to limit superior and limit inferior

In Integration: Riemann integrals use supremum and infimum of partitions

In Functional Analysis: Norms and operator theory rely on supremum

In Optimization: Finding optimal values often means finding supremum/infimum

In Measure Theory: Outer measure uses infimum of coverings

Understanding supremum and infimum is essential for rigorous analysis. They formalize the idea of "best possible bound" and are fundamental to the completeness of real numbers!

My Teaching Experience

I always tell students: think of supremum as "trying to be the maximum, even if it can't quite make it." The supremum is as close as you can get to a maximum while still being an upper bound.

The completeness axiom—that every bounded set has a supremum—is what separates ℝ from ℚ. It's the reason calculus works on real numbers!

I've recently started a new series on my YouTube channel "Maths Mastery with Dr. Upasana P Taneja" focused on Topology of Real Numbers, specially designed for B.Sc. 2nd year students! In this series, I work through many examples showing how to find and verify supremum and infimum, along with other fundamental concepts. These concepts appear everywhere in advanced analysis and are crucial for your exams!

The beauty of supremum and infimum lies in their precision: they capture the exact boundary of a set, even when that boundary isn't reached by any element in the set. That's the elegance of real analysis!

Finding the best bound—that's the supremum way!
Dr. Upasana P Taneja

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