1 Definition and basic ideas
The infimum of a set is the greatest number that does not exceed any element of the set. In ordinary analysis, this notion captures the best possible lower estimate for a collection of numbers. When an infimum exists, it summarizes the lower extent of the set in a single value and plays a central role in order theory, real analysis, and optimization.
1.1 Lower bounds
A lower bound of a set is any number that is less than or equal to every element of that set. A set may have many lower bounds. For example, any number less than or equal to all members of an interval is a lower bound, and all smaller numbers are lower bounds as well. Lower bounds form the starting point for defining the infimum.
1.2 Greatest lower bound
Among all lower bounds, the infimum is the greatest one. It is the tightest lower estimate available. If a set has a lower bound that no other lower bound exceeds, then that number is the infimum. This formulation is often called the greatest lower bound property.
1.3 Infimum versus minimum
A minimum is an element of the set that is smaller than or equal to every other element. An infimum need not belong to the set. When the infimum is contained in the set, it is also the minimum. When it is not contained, it still describes the lowest boundary of the set but is not itself attained by any member.
1.4 Notation and terminology
The infimum is commonly written as \(\inf\). For a set \(A\), one writes \(\inf A\). In some texts, especially in order theory, the term greatest lower bound is used alongside infimum. The notation emphasizes that the value depends on the entire set rather than on an individual element.
2 Existence and uniqueness
Whether an infimum exists depends on the surrounding ordered structure. In the real numbers, bounded-below nonempty sets always have an infimum. In more general ordered sets, this is not automatic and may require additional completeness assumptions.
2.1 Infimum in the real numbers
The real numbers are complete in the sense that every nonempty set bounded below has a greatest lower bound. This completeness principle makes the infimum a reliable tool in analysis. It allows one to pass from a collection of lower bounds to a single distinguished real number.
2.2 Uniqueness of the infimum
If an infimum exists, it is unique. Two different numbers cannot both be the greatest lower bound of the same set, because each would have to be at least as large as the other. This uniqueness is a basic consequence of the order relation.
2.3 Bounded below sets
A set is bounded below if it has at least one lower bound. Such sets are the natural domain for infimum questions. The presence of a lower bound suggests that the set has a floor, even if that floor is not attained by any element.
2.3.1 Nonempty bounded sets
For nonempty subsets of the real numbers that are bounded below, the infimum exists. This is one of the fundamental consequences of completeness. It fails in some smaller ordered systems, where a bounded set may have lower bounds without a greatest one.
2.3.2 Unbounded sets
If a set is not bounded below, then it has no real infimum. In extended settings one may assign the value \(-\infty\), but within the ordinary real numbers no greatest lower bound can be formed. Such sets extend indefinitely downward.
3 Characterizations of the infimum
The infimum can be described in several equivalent ways. These characterizations are useful in proofs, because one description may be easier to apply than another depending on the problem.
3.1 Order-theoretic characterization
Order-theoretically, \(a=\inf A\) if \(a\) is a lower bound of \(A\) and every lower bound of \(A\) is less than or equal to \(a\). This expresses the defining maximality of the lower bound directly in terms of the order relation.
3.2 Epsilon characterization
A number \(a\) is the infimum of a set \(A\) if, for every positive \(\varepsilon\), there exists an element \(x\in A\) with \(x<a+\varepsilon\), while still \(a\le x\) for all \(x\in A\). This means the set has elements arbitrarily close to the infimum from above.
3.3 Characterization by lower bounds
The infimum is the largest element of the set of all lower bounds. This viewpoint shifts attention from the original set to its collection of admissible lower estimates. The infimum is then the endpoint of that lower-bound collection.
3.4 Characterization via approximation from above
Another way to describe the infimum is through approximation from above: values in the set can be found as close as desired to the infimum without crossing below it. This is especially useful in analysis, where one often estimates quantities by descending sequences of upper approximations.
4 Examples
Examples show how the infimum behaves in concrete situations. They illustrate the difference between a boundary value that is attained and one that is only approached.
4.1 Finite sets
For a finite nonempty set of real numbers, the infimum is simply the smallest element, so it is also the minimum. Finite sets therefore provide the most elementary case, with no distinction between greatest lower bound and least member.
4.2 Open intervals
For the open interval \((a,b)\), the infimum is \(a\), even though \(a\) is not contained in the interval. The set approaches \(a\) from above but never reaches it. This is a standard example of an infimum that is not a minimum.
4.3 Closed intervals
For the closed interval \([a,b]\), the infimum is \(a\), and here it is also the minimum. Because the endpoint is included, the boundary value is attained. This makes closed intervals especially simple to analyze.
4.4 Sets without a minimum
A set may have an infimum but no minimum. The positive real numbers are a familiar example: they are bounded below by 0, and their infimum is 0, yet 0 is not a positive number and therefore not a member of the set. The lower edge exists without being realized internally.
4.5 Sets with infimum outside the set
More generally, any set that accumulates toward a lower limit without containing it has an infimum outside the set. Such behavior is common in intervals, sequences, and solution sets in analysis. The infimum records the limiting boundary rather than an actual member.
5 Properties
Infimums obey several structural rules that make them useful in proofs and calculations. These properties reflect the way lower bounds interact with inclusion and set operations.
5.1 Relation to set inclusion
If one set is contained in another, its infimum cannot be smaller than the infimum of the larger set, provided both are bounded below. A smaller set has fewer elements to constrain it, so its greatest lower bound may move upward or remain the same. Inclusion therefore reverses the intuitive size comparison in the lower-bound direction.
5.2 Behavior under unions and intersections
For unions, the infimum is generally the smaller of the infima only when the relevant sets are both bounded below in a compatible way. For intersections, the infimum may increase because the intersection imposes more restrictions. These relationships depend on the geometry of the sets and are not as simple as ordinary arithmetic rules.
5.3 Monotonicity
When a family of sets expands, its infimum cannot increase unless the added elements force the lower bound downward. More precisely, if \(A\subseteq B\), then \(\inf B\le \inf A\). This monotonic behavior is a direct consequence of the definition of lower bound.
5.4 Infimum of subsets
A subset may have a different infimum from its parent set. Removing elements can raise the infimum, while adding smaller elements can lower it. This sensitivity makes the infimum useful for tracking how sets change under restriction.
5.5 Infimum and boundedness
The existence of an infimum implies boundedness below. In the real numbers, the converse holds for nonempty sets. Thus infimum and lower boundedness are tightly linked, with completeness bridging the gap between them.
6 Relationship to the supremum
The infimum and supremum are dual concepts. One concerns the greatest lower boundary, the other the least upper boundary. Together they provide complementary descriptions of the extent of a set.
6.1 Duality between infimum and supremum
Many statements about infimum have mirror-image versions for supremum. Replacing “lower” with “upper,” and reversing inequalities, transforms one concept into the other. This duality is a recurring theme in order theory.
6.2 Infimum as negative supremum
For a set \(A\) of real numbers, \(\inf A\) can be expressed as the negative of the supremum of the negated set: \(\inf A=-\sup(-A)\). This identity allows results about suprema to be transferred to infima through sign reversal.
6.3 Shared properties and differences
Both infimum and supremum may or may not belong to the set. Both are unique when they exist. Their main difference is directional: the infimum controls the lower edge, while the supremum controls the upper edge. Together they determine the range of a bounded set.
7 Infimum in different mathematical settings
The notion of infimum extends beyond the real numbers. In more abstract settings, it is defined relative to an order relation, and its existence depends on structural properties of the space.
7.1 Ordered sets and lattices
In a partially ordered set, an infimum is the greatest lower bound when it exists. In lattice theory, every pair of elements may have an infimum, called their meet. This abstraction allows the concept to apply to algebraic and combinatorial structures as well as numbers.
7.2 Extended real numbers
In the extended real line, one may include \(-\infty\) and \(+\infty\) so that every subset has an infimum and supremum in a broader sense. This convention is useful in analysis and optimization, especially when dealing with unbounded functions or feasible regions.
7.3 Metric and topological contexts
Although infimum is fundamentally an order concept, it often appears in metric and topological arguments through distance, closure, and limit processes. For example, the infimum of distances from a point to a set defines the distance to that set. This connects order structure with geometry.
7.4 Complete lattices
In a complete lattice, every subset has both an infimum and a supremum. Such structures provide the broadest natural environment for order-based extremal arguments. Many results that are delicate in the real numbers become formally straightforward in complete lattices.
8 Computation and application
Infimums are not merely abstract objects; they are frequently computed and applied in analysis, optimization, and the study of limits. They help formalize best possible lower estimates in many problems.
8.1 Finding infima of real sets
To find an infimum, one usually identifies all lower bounds and then determines the largest among them. For intervals and simple algebraic sets, this can often be done by inspection. In more complicated cases, one may use inequalities, monotonicity, or limiting sequences.
8.2 Optimization problems
In optimization, the infimum represents the best achievable objective value, even when no minimizing point exists. This distinction is important in constrained problems, where a cost may approach a limit without attaining it. The infimum therefore describes the ideal outcome of minimization.
8.3 Limits and convergence
Infimums are used to define and analyze limits of sequences of sets, functions, and numerical expressions. They often appear in constructions involving bounds and envelopes. In convergence arguments, the infimum can capture a limiting threshold from below.
8.4 Variational methods
Variational methods frequently search for an infimum of an energy functional. The value may be used to prove existence of minimizers or to show that an approximate minimizing sequence is well behaved. In this setting, the infimum is a central object because it formalizes the least possible energy compatible with the problem.