Somehow or another I cannot manage to retain the properties of the limit superior in my head for more than ten minutes. This has been plauging me for years. I am writing this blog to help me understand and remember these operations, and their significance in applied mathematics.
A general sequence may accumulate to a certain point, or many points, or no points. The limit superior is the largest point about which a sequence accumulates around. The limit superior is the smallest point about which a sequence accumulates.
Now, the limit superior is not necessarily equal to the largest point in the sequence. The limit superior may not exist in the sequence. Furthermore there may be points in the sequence that are larger than the limit superior. This is expounded next.
The limit superior is not necessarilty equal to the largest point in the sequence. The limit superior is the largest point which a sequence accumulates around. That is, it is the largest of all points that occur infinitely often in the sequence. There may be larger values than the limit superior in the sequence, but only finitely many of these values.
The above exposition is imprecise. Technically, there may be infinitly many terms greater than the limit superior. However, within an arbitrarily small distance above the limit superior, there can be only finitely many points. So we can say that the series does not stay significantly above the limit superior for ever — eventually, one will fall close to it in a certain sense. The sense depends on context.
The converse is true for the limit inferior.
The limit superior and limit inferior are quite closely related. For instance, in set theory, the complement of the limit superior of a sequence is equivalent to the limit inferior of the complement series. For real numbers, the limit superior of a negated sequence is equivalent to the negated limit inferior. There is a sort of symmetry to these operations.
The limit superior and limit inferior can be used to define limits. The limit of a sequence can be defined as the value of the limit superior or the limit inferior when the limit superior and the limit inferior are equivalent. Where the limit superior and limit inferior are not equal, the limit does not exist.
The limit superior and limit inferior can conveniently be defined in terms of tail sequences. Consider an arbitrary sequence. Now throw away the first n terms of the sequence. The remaining sequence is the n-th tail sequence, or the remaining terms of the original sequence.
Now the limit superior can be defined with tail sequences. The limit superior is the limit (as n grows large) of the supremum value of the n-th tail sequence. So as the tail sequence shrinks, each sequence will have some sort of supremum, or upper bound. The limit of these values is the limit supremum.
The sequences of the supremum's is strictly decreasing. Therefore, the limit of this sequence is equivalent to the infimum of the sequence. The limit superior is therefore the infimum of the tail sequences's supremums.
The converse is true of the limit inferior as well. The limit inferior is defined as the supremum of the tail sequence's infimums.
The limit superior may not be defined. This is a technical point, which I am still learning about. But for sequences of real numbers, if the underlying space is not complete, the limit superior may not exist. For sequences of sets, if the space is not a complete lattice, then the limit superior may not exist. From what I currently understand, some notion of completeness needs to be defined on the underlying space for the limit superior and limit inferior to exist. I think that this is why analysts will introduce the extended real numbers (as opposed to just working with the real numbers).
The properties of limit superior and limit inferior result in some neat properties that I will briefly state.
* The details of completeness are not completely clear to me currently. Although I understand completeness of the real numbers, and analogous ideas of complete lattices are clear, I do not understand how these ideas generalize to arbitrary sequences yet, or how it connects to the idea of order. This is something to look into in the future.