Many undergraduates have difficulty understanding the notions of the lim inf and lim sup of a sequence.
[The full names for these appear to vary from author to author. In the first version of this post I called them “limit infimum” and “limit supremum”, but I think that “limit inferior” and “limit superior” are the most widely accepted names. ]
Of course, there is the basic problem that students confuse with (which is, strictly speaking, meaningless, but might generously be interpreted as meaning one of or ). However, what I really mean is that the students often fail to grasp what and really mean. (See below for some more details of what I mean by this!)
As with epsilon and delta, we may be tempted to avoid confronting the students’ difficulties with lim inf and lim sup. For example, we can often choose between using lim inf and lim sup or using the sandwich theorem (also known as the squeeze rule). A typical example of this is the standard exercise where you have to prove the following fact at the start of a course on metric spaces.
Let be a metric space, and let and be convergent sequences in with limits and respectively. Prove that as .
I leave it to the reader to supply two proofs, one using the sandwich theorem, and another using and .
As with epsilon and delta, it may be that postponing discussion of and , or avoiding them altogether, is not in the best interests of the student. I have to admit that I am not sure! But I think it is worth investigating possible ways to help students to understand and .
In the following, for convenience, we work in the extended real line
This is convenient, because every subset of has a supremum and an infimum in : there is no need to worry about boundedness and non-emptiness. Those who prefer to work in should add in appropriate assumptions below where necessary.
The standard approach to lim inf and lim sup
The following approach to lim inf and lim sup is entirely standard.
Let be a sequence in . Then it is standard to define sequences and in as follows: for each ,
We may then define
Once you have decided on an appropriate definition of convergence in , you can confirm that we also have
These definitions are very clean, and are easy to apply, e.g., to prove results in the theory of measure and integration. But they do not, in themsleves, give the student a very good idea of what and really mean for a typical sequence. In my opinion, even calculating the and of a few examples does not really help as much as you would expect.
One approach that can help a little is to explain that is the minimum of all the possible limits of subsequences of the sequence , and similarly for , with in place of .
However, in my opinion, what we should try to get across is what and tell us about where can actually be as becomes large.
The absorption approach to lim inf and lim sup
Let be a sequence of extended real numbers. Set and . For the rest of this post, , and will be fixed.
What I think we would like students to understand is that, for large , is “almost” in , and there is no strictly smaller closed interval for which this is true.
Recall, in my terminology, a set absorbs a sequence if at most finitely many terms of the sequence lie outside the set.
In terms of absorption we can say various things about the relationships between , and . These standard facts are usually expressed using more standard terminology, e.g., in terms of a sequence “eventually lying within a set” or, for non-absorption, infinitely many terms of the sequence lying outside a set.
Let and be extended real numbers.
- If , then absorbs the sequence.
- If , then does not absorb .
- If , then absorbs .
- If , then does not absorb .
- If then absorbs .
- If is a proper subset of , then does not absorb .
Of course, we do not know whether or not one or both of and absorb . However, it is true that is equal to the intersection of all closed extended-real intervals which absorb the sequence .
Note added 4/12/09: Note that condition 6 in the above list
is not strong enough to recover much information in situations where itself fails to absorb the sequence. For example,for the sequence , the interval satisfies conditions 5 and 6 above, without being equal to .
As mentioned above, all of these statements may be expressed using more standard terminology. Is the language of absorption helpful here?
In due course, I plan to return to this topic in the setting of function limits rather than sequences. This will then connect up with continuity and semicontinuity of functions.