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Infimum — Definition, Formula & Examples

The infimum of a set is the greatest lower bound — the largest value that is less than or equal to every element in the set. Unlike the minimum, the infimum does not need to belong to the set itself.

Let SS be a nonempty subset of R\mathbb{R} that is bounded below. The infimum of SS, written inf⁡(S)\inf(S), is the unique real number mm such that (i) m≤sm \leq s for every s∈Ss \in S, and (ii) if m′m' is any other lower bound of SS, then m′≤mm' \leq m.

Key Formula

inf⁡(S)=max⁡{ m∈R:m≤s for all s∈S }\inf(S) = \max\{\, m \in \mathbb{R} : m \leq s \text{ for all } s \in S \,\}
Where:
  • SS = A nonempty subset of the real numbers that is bounded below
  • mm = A lower bound of S; the infimum is the greatest such m
  • ss = An arbitrary element of S

How It Works

To find the infimum of a set, first identify all lower bounds — values that sit at or below every element. Then pick the largest among those lower bounds; that is the infimum. If the set contains its smallest element, the infimum equals that minimum. If the set approaches a boundary without reaching it (as with an open interval), the infimum exists but is not an element of the set. By the completeness property of R\mathbb{R}, every nonempty set that is bounded below has a real-valued infimum.

Worked Example

Problem: Find the infimum of the open interval S=(2,7)S = (2, 7).
Step 1: Identify lower bounds. Any number m≤2m \leq 2 satisfies m≤sm \leq s for all s∈(2,7)s \in (2, 7), so the set of lower bounds is (−∞,2](-\infty, 2].
m≤2  ⟹  m≤s for all s∈(2,7)m \leq 2 \implies m \leq s \text{ for all } s \in (2,7)
Step 2: Choose the greatest lower bound. The largest value in (−∞,2](-\infty, 2] is 22.
inf⁡(S)=2\inf(S) = 2
Step 3: Note that 2∉(2,7)2 \notin (2, 7), so the set has no minimum. The infimum exists even though it is not an element of SS.
Answer: inf⁡(2,7)=2\inf(2,7) = 2. The infimum is 2, but it is not a minimum because 2 does not belong to the open interval.

Why It Matters

The infimum is central to real analysis: it underpins the formal definitions of limits, continuity, and the Riemann integral. In optimization and economics, identifying greatest lower bounds lets you characterize best-possible worst-case outcomes even when no exact minimum is attained.

Common Mistakes

Mistake: Assuming the infimum must be an element of the set (confusing infimum with minimum).
Correction: The infimum is the greatest lower bound and may lie outside the set. It equals the minimum only when that minimum exists. For example, inf⁡(0,1)=0\inf(0,1) = 0, but 0∉(0,1)0 \notin (0,1).