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Supremum — Definition, Formula & Examples

The supremum of a set is the smallest value that is greater than or equal to every element in the set. It is often called the least upper bound (abbreviated lub or sup).

Let SS be a nonempty subset of R\mathbb{R} that is bounded above. A real number M=sup⁡SM = \sup S is the supremum of SS if (1) MM is an upper bound of SS, meaning x≤Mx \leq M for all x∈Sx \in S, and (2) no number less than MM is an upper bound of SS, meaning for every ϵ>0\epsilon > 0 there exists some x∈Sx \in S with x>M−ϵx > M - \epsilon.

Key Formula

sup⁡S=min⁡{ M∈R:x≤M for all x∈S }\sup S = \min\{\, M \in \mathbb{R} : x \leq M \text{ for all } x \in S \,\}
Where:
  • SS = A nonempty subset of the real numbers that is bounded above
  • MM = An upper bound of S
  • sup⁡S\sup S = The least (smallest) upper bound of S

How It Works

To find sup⁡S\sup S, first confirm the set is bounded above. Then identify the smallest upper bound. If the set contains its largest element, the supremum equals that maximum. If not, the supremum is a limit point that the set approaches but never reaches. For example, the open interval (0,1)(0, 1) has no maximum element, yet sup⁡(0,1)=1\sup(0,1) = 1 because 1 is the tightest upper bound.

Worked Example

Problem: Find the supremum of the set S={ 1−1n:n∈N }={0,  12,  23,  34,  …}S = \left\{\, 1 - \dfrac{1}{n} : n \in \mathbb{N} \,\right\} = \left\{0,\; \dfrac{1}{2},\; \dfrac{2}{3},\; \dfrac{3}{4},\; \ldots\right\}.
Step 1: Check that S is bounded above. Every element has the form 1−1/n<11 - 1/n < 1, so 1 is an upper bound.
1−1n<1for all n∈N1 - \frac{1}{n} < 1 \quad \text{for all } n \in \mathbb{N}
Step 2: Show no number less than 1 is an upper bound. Pick any candidate M=1−ϵM = 1 - \epsilon with ϵ>0\epsilon > 0. Choose nn large enough so that 1/n<ϵ1/n < \epsilon. Then 1−1/n>1−ϵ=M1 - 1/n > 1 - \epsilon = M, so MM fails to be an upper bound.
n>1ϵ  ⟹  1−1n>1−ϵn > \frac{1}{\epsilon} \implies 1 - \frac{1}{n} > 1 - \epsilon
Step 3: Since 1 is an upper bound and no smaller number is, the supremum is 1. Note that 1 itself is not in S, so S has no maximum.
sup⁡S=1\sup S = 1
Answer: sup⁡S=1\sup S = 1. The set gets arbitrarily close to 1 but never reaches it, so the supremum exists even though the maximum does not.

Why It Matters

The Least Upper Bound Property — that every nonempty, bounded-above subset of R\mathbb{R} has a supremum in R\mathbb{R} — is the axiom that distinguishes the reals from the rationals. Proofs of the Intermediate Value Theorem, the Bolzano–Weierstrass Theorem, and the convergence of bounded monotone sequences all rely on it. You will encounter the supremum constantly in real analysis, measure theory, and functional analysis courses.

Common Mistakes

Mistake: Assuming the supremum must belong to the set (confusing sup with max).
Correction: A maximum is a supremum that is also an element of the set. Many sets (like open intervals or sequences approaching a limit) have a supremum that lies outside the set. Always check membership separately.