Intuitions: limit supremum and limit infimum of sets, sequences and functions

7 min readOct 7, 2022

Introduction

I decided to write this article because I noticed a lack of intuitive clarity regarding the concepts of limit supremum (also called limit superior) and limit infimum (also called limit inferior) among many. Students of mathematics, statistics or data science are often able to write their mathematical definitions and give one or two examples. But not all of them really understand them intuitively.

Here, I try to give clear intuitions about limit supremum and limit infimum in various contexts all in one place using more words than mathematical symbols. I do not include examples as those are easily available in textbooks. I include what is generally not found in textbooks — linguistic expression of the definitions and rough visualizations which often help more than symbols to form intuition.

Limit supremum or limit infimum can be considered: 1) for sets, 2) for sequences of real numbers, 3) for sequences of sets, 4) for functions, and 5) for sequences of functions. Although the fundamental ideas are the same, one needs to translate these ideas properly for each different context.

At the outset, note that the ideas of limit supremum or limit infimum arise in mathematical analysis, or simply analysis for brevity. Analysis basically studies infinities. The presence of infinite number of something, be it numbers, or sets, or functions is entertained and examined in analysis. So, many concepts of analysis, although defined for finitely many elements as well, will only be useful in the presence of infinitely many elements. Limit superior and limit inferior are no exceptions. For them to exist in any context, there must be infinite number of something present. For finite number of something, they do not exist be definition.

With this understanding, let us take up the above five contexts one by one.

1) Limit supremum/infimum of sets

[Definitions in words]

The limit supremum of a set is the supremum of the limit points of the set.¹
The limit infimum of a set is the infimum of the limit points of the set.¹ (Of course, the set has to be an ordered set, i.e., a set where elements can be ordered.)

Intuition :: I presume the reader somewhat knows what a limit point is. A point (of the same type as the elements of the set) is a limit point or a cluster point or an accumulation point of a set if any neighborhood of that point contains an element of that set other than the point itself.

However, this means, within any neighborhood of a limit point, we can get further smaller neighborhoods containing an element of that set other than the limit point. So basically, any neighborhood of a limit point contains infinitely many elements of the set (see illustration). Thus, for a set to have limit points, it must have infinitely many elements to begin with.

Simplistic illustration of limit points for intuition. Consider a set with infinite points (black dots). A and B are limit points of the set as any neighborhood of them includes infinitely many points of the set. But C is not a limit point (handsketched by the author).

However, a limit point itself may not be an element of the set. For this, it is generally being referred to as a point, not an element. Limit points are points around which a set accumulates or clusters around or gravitates.

The largest and the smallest of such limit points are the limit supremum and the limit infimum of a set.

2) Limit supremum/infimum of real sequences

[Definitions in words]

The limit supremum of a sequence of real numbers is the smallest real number that is greater than or equal to infinitely many members of the sequence. (Supremum is the least of all upper bounds of the whole sequence, limit supremum is the least of all upper bounds of infinitely many members of the sequence [upper bounds of only finitely many members don’t count.])
The limit infimum of a sequence of real numbers is the largest real number that is smaller than or equal to infinitely many members of the sequence. (Infimum is the greatest of all lower bounds of the whole sequence, limit infimum is the greatest of all lower bounds of infinitely many members of the sequence [lower bounds of only finitely many members don’t count.])

Intuition :: To be a candidate for the limit supremum, a number has to be greater than or equal to infinitely many members of the sequence.

For this, one generally considers the tails of the sequence (see illustration). Why tails? Why not heads? Because given any member, all the members to the left of it (head) are finite in number as the sequence starts on the left and the starting point is known. ([Avoiding bi-infinite sequences for now] a sequence is a mapping from the set of natural numbers which starts at 1, so the starting side is known. You can tell the first natural number, the first 2 natural numbers, etc. but cannot tell the last 2 natural numbers.) Only the right side is infinite. So, where there is a talk of infinitely many members, you have to consider a tail.

Illustration for limit supremum and limit infimum of a sequence of numbers (handwritten by the author)

Any tail of the sequence is infinitely long. Hence, the supremum of any tail is greater than or equal to infinitely many members of the sequence. Thus, the supremum of any tail is a candidate for limit supremum.

Further, as successive tails are subsets of the previous ones, the corresponding supremums form a monotonically decreasing sequence of candidates for the limit supremum. However, the infimum (= limit, in this case) of this sequence of candidates is the smallest number that already bounds infinitely many members of the original sequence from above–which is precisely the purpose. So, we reject all larger candidates and this infimum, as the best/most parsimonious candidate, is elected as the limit supremum of the original sequence. Similarly, limit infimum can be understood.

[Mathematical definitions]

Guided by this intuition, now let us look at the mathematical definitions.²

Mathematical definition of limit supremum and limit infimum for a real sequence (from author’s own note)

3) Limit supremum/infimum of sequences of sets

[Definitions in words]

The limit supremum of a sequence of sets is the intersection of all sets that can be formed by a countably infinite union of sets from the sequence.
The limit infimum of a sequence of sets is the union of all sets that can be formed by a countably infinite intersection of sets from the sequence.

The above definitions are general definitions. In the context of probability theory, the definitions become more specific. Consider a probability space consisting of a sample space, a sigma field over the sample space, and the probability measure. Each element of the sample space is an outcome of a random experiment. Each element of the sigma field is an event which is a set of such outcomes. So, a sequence of events is just a sequence of sets. In this setting, limit supremum and limit infimum can be expressed in terms of outcomes as follows.

[Definitions in words for a probability space]

The limit supremum of a sequence of events is the collection of all outcomes that occur in the sequence ‘infinitely often’ or ‘frequently’, i.e., that never leave the sequence ‘for good’.³
The limit infimum of a sequence of events is the collection of all outcomes that occur in the sequence ‘eventually’ or ‘ultimately’, i.e., that from some point onward stays in the sequence ‘for good’.³

[Mathematical definitions]

With this intuition, now let us look at the mathematical definitions.³

Mathematical definition of limit supremum and limit infimum for a sequence of sets (from author’s own note)

See that the mathematical definition of limit supremum embodies an eliminative process (see illustration) whereby from an initial set (the first union with n=1), already containing the elements of the limit supremum, finitely-occurring elements are successively eliminated (by taking intersections) till such elements are there. What are left are elements that occur infinitely often.

Illustration for limit supremum and limit infimum of a sequence of sets (handsketched by the author)

The definition of the limit infimum embodies a constructive process whereby the set of the limit infimum is constructed by successively adding (through union) infinitely-occurring elements (intersection of an infinite number of sets) till such elements are there.

4) Limit supremum/infimum of functions

[Definitions in words]

The limit supremum of a function at a given point is the limit (=infimum, in this case) of the supremums of the function over successively decreasing neighborhoods of the point excluding the point itself.
The limit infimum of a function at a given point is the limit (=supremum, in this case) of the infimums of the function over successively decreasing neighborhoods of the point excluding the point itself.

See this illustration.

Illustration for limit supremum and limit infimum of a function (handsketched by the author)

Now, let us look at the mathematical definitions.⁴

[Mathematical definitions]

Mathematical definition of limit supremum and limit infimum for a function (from author’s own note)

5) Limit supremum/infimum of sequences of functions

[Definitions in words]

The (pointwise) limit supremum of a sequence of functions is a function whose value, for a given value of the argument, is equal to the limit supremum of the sequence consisting of the values of the individual functions at that value of the argument.
The (pointwise) limit infimum of a sequence of functions is a function whose value, for a given value of the argument, is equal to the limit infimum of the sequence consisting of the values of the individual functions at that value of the argument.

See this illustration.