Supremum and Infimum: Definitions, Examples and Key

The supremum (sup) of a set is its least upper bound — the smallest ceiling that sits above every member. The infimum (inf) is the greatest lower bound — the largest floor beneath every member. Unlike a maximum or minimum, the supremum and infimum do not have to belong to the set.

This matters because many important sets have no biggest or smallest element — the open interval (0, 1) has no maximum — yet their sup and inf still exist and pin down the set's edges exactly. Analysis uses sup and inf wherever max and min fail to exist.

Key Concepts

• Upper bound: any number M with x ≤ M for all x in the set.
• Supremum: the least of all upper bounds; if the set has a maximum, sup equals it.
• Infimum: the greatest of all lower bounds; if the set has a minimum, inf equals it.
• Supremum need not be a member: sup of (0, 1) is 1, but 1 ∉ (0, 1).
• Existence: completeness guarantees sup and inf exist for non-empty bounded sets of reals.

Formulas

• Supremum test: M = sup S exactly when (i) x ≤ M for all x ∈ S, and (ii) for every ε > 0 there is x ∈ S with M − ε < x.
• Infimum test: m = inf S exactly when (i) x ≥ m for all x ∈ S, and (ii) for every ε > 0 there is x ∈ S with x < m + ε.
• ε-approximation: the supremum can be approached arbitrarily closely by members of the set.
• Monotone link: an increasing bounded sequence converges to the supremum of its terms.

Worked Example

Let S = {1/n : n ∈ ℕ} = {1, 1/2, 1/3, 1/4, …}. The largest member is 1 (when n = 1), so sup S = 1, and since 1 ∈ S it is also the maximum. For the infimum: every term is positive, so 0 is a lower bound; −1 and −5 are also lower bounds, but 0 is the greatest. Is 0 in S? No — 1/n > 0 for every n, so S has no minimum, yet inf S = 0. Verify with the ε-test: take ε = 0.001; we need a member smaller than 0 + 0.001. Choose n = 1001: 1/1001 ≈ 0.000999 < 0.001, and it belongs to S. Since such a member exists for every ε > 0, 0 is indeed the greatest lower bound. The set is squeezed between inf 0 (outside the set) and sup 1 (inside the set).

Common Misconceptions

• The supremum must belong to the set. Not so — sup (0, 1) = 1 lies outside the set; only closed sets contain their sup.
• Every set has a supremum. Unbounded sets like ℕ have no upper bound at all, so no supremum exists.
• Supremum and maximum are the same word. The maximum is a member; the supremum is a bound. They coincide only when the bound is attained.

Real-World Applications

• Optimisation: the least upper bound of achievable values defines the best possible outcome.
• Integration: upper and lower sums squeeze the integral between sup and inf approximations.
• Probability: sup of probabilities over events appears in measure theory.

Supremum and infimum capture the exact edges of a set even when no largest or smallest member exists. They are the precise tools that make "approach but never reach" rigorous.