Can an infimum also be a minimum?

Yes, when the greatest lower bound belongs to the set.

Can a set have two different infima?

No. If two numbers were both greatest lower bounds, each would have to be at least the other, so they would be equal.

Does a decreasing sequence always reach its infimum?

No. The worked sequence approaches 3 but never reaches it at a finite index.

AI Glossary / Mathematics and optimization / Intermediate

Infimum (or greatest lower bound)

Definition

The infimum of a nonempty set of real numbers bounded below is its greatest lower bound: the largest number that is no greater than every member of the set. Unlike a minimum, an infimum need not belong to the set. In optimization, this distinction separates the best limiting objective value from a value actually achieved by a feasible decision.

What it means

Suppose a collection of attainable costs gets closer and closer to a boundary without ever reaching it. That boundary can still be a precise mathematical answer to the question of how low the costs can approach. It is not necessarily an attainable cost.
A lower bound sits at or below every value in the set. Many lower bounds can exist; the infimum is the tightest one. If it is itself in the set, it is also the minimum. The word greatest describes its position among lower bounds, not among the values whose lower boundary is being studied.

How it works

To establish that a number L is the infimum, verify two things. First, every value in the set is at least L. Second, values in the set can be found arbitrarily close to L from above: for every positive epsilon, some set member is less than L+epsilon. The second condition rules out any larger lower bound.
For an objective f over feasible set X, write inf over x in X of f(x) for the lower boundary of attainable objective values. A minimizer is a feasible x that attains it. Existence of the infimum as a real number does not by itself establish existence of a minimizer.

Three practical examples

An optimization analyst can distinguish a proven lower objective bound from the cost of the best feasible solution found.
A machine-learning researcher can describe a loss approaching a limit while model parameters continue changing without reaching a finite optimizer.
A mathematical modeler can detect when a strict feasibility requirement prevents an otherwise natural boundary solution from being allowed.

Think of it this way

Think of descending toward the bottom of an open stairwell whose steps approach a marked height without reaching it. The mark describes the lower limit of the step heights, but it is not a step you can stand on. The analogy is limited because a mathematical set need not consist of a sequence or resemble a physical staircase.

Worked example

Consider the original set S={3+2/n : n is a positive integer}. Its first values are 5, 4, 11/3, 3.5, and 3.4. Every member is greater than 3, so 3 is a lower bound. To approach within any chosen epsilon>0, select n>2/epsilon. Then 2/n<epsilon and 3+2/n<3+epsilon. Thus inf S=3.
There is no minimum: for any positive n, choosing n+1 produces a strictly smaller member. The number 3 is not in S because 2/n never equals zero for a finite positive n.
Suppose a report requires a value strictly less than 3.01. The inequality 3+2/n<3.01 requires n>200, so n=201 qualifies. Choosing n=200 gives exactly 3.01 and fails the strict requirement. This also demonstrates that approaching the infimum to a tolerance is different from attaining it.
Now add the point 3 to form S-plus=S union {3}. The infimum stays 3, but the set now has a minimum of 3.
For an optimization contrast, minimize f(x)=x over x>0. Its infimum is zero and no feasible minimizer exists. Replacing the strict constraint with x≥0 produces a minimum at x=0. A solver returning a very small positive number in the first problem has found an approximate value, not a proof that a feasible zero exists.

Illustrative case study

In a fictional example, Elm Routing prototypes an optimization model with a strictly positive reserve variable. Repeated runs return smaller reserves and slightly improved objective values. A manager assumes the solver is failing to locate the exact best setting.
An analyst inspects the mathematical domain and shows that the simplified model has a limiting objective value without an attainable optimum. The team revisits what the reserve requirement is meant to express and chooses a meaningful positive operational bound. This invented case illustrates a modeling issue, not a claim that real routing systems necessarily have unattained optima.

Common misconceptions and mistakes

Do not confuse the infimum with the smallest value observed in a finite sample. A finite nonempty sample has a minimum, but an unobserved feasible value may be smaller. Finding a very low value supplies an upper bound on a minimization problem's infimum, not a lower-bound certificate.
Not every set has a finite real infimum. A nonempty set unbounded below has no real lower bound; extended-real notation often writes its infimum as negative infinity. Some optimization conventions also assign positive infinity to the infimum of an empty feasible set. State the convention when these cases arise.
An infimum is a value, whereas an argmin describes the feasible points attaining a minimum. The argmin can be empty even when the infimum is finite.

Business use

Understanding infima helps teams interpret optimization results and avoid promising unattainable targets. A useful report distinguishes the best feasible value, any valid lower bound, the gap between them, and whether attainment has been established.
For a business model, an unattained boundary may reveal a missing practical constraint or an unrealistic idealization. It is not automatically a reason to discard the model. Approximate decisions can be useful when tolerances and operating requirements are explicit. The crucial step is to explain what the reported value represents and whether there is a feasible decision that actually delivers it.

Check your understanding

For the interval (2,7], what are the infimum and minimum?

Answer: The infimum is 2. There is no minimum, because 2 is excluded and every allowed number greater than 2 has a smaller allowed number between it and 2.

Related terms

Supremum (or least upper bound)
Convergence
minimum
lower bound
supremum
maximum
feasible set
objective value
argmin
optimality gap

Learn next

Compare the infima and minima of (0,1), [0,1), and the positive integers. For each, identify a lower bound that is not the greatest lower bound. Then practice the two-part proof: establish a bound and show that no larger number can remain a bound.

Sources

Primary terminology source: Alexander Jung and collaborators, The Aalto Dictionary of Machine Learning, infimum entry: https://aaltodictionaryofml.github.io/ADictML_Main.pdf . The greatest-lower-bound definition and attainment distinction were consulted. The sequence, tolerance calculation, and fictional modeling case are original illustrations.

Previous: Inferential theory of learning Next: Infinite horizon