Lim sup and Lim inf

Convergent sequences are useful objects, but the unfortunate truth is that most sequences do not converge.  Nevertheless, we would like to have a language for discussing the asymptotic behavior of any real sequence {a_j} as j -> infinity.  That is the purpose of the concept of "limit superior" (or "upper limit") and "limit inferior" (or "lower limit").

DEFINITION 1      Let {a_j}be a sequence of real numbers. For each j let 

                                               A_j  = inf {a_j, a_j+1 ,  a_j+2 , ...}

Then {A_j} is a monotone increasing sequence (since as j becomes large we are taking the infimum of a smaller set of numbers), so it has a limit.  We define the limit inferior of {a_j} to be 

                             lim inf a_j     =    lim A_j         as j -> infinity

Likewise, let 

                                               B_j  = inf {a_j, a_j+1 ,  a_j+2 , ...}

Then  {B_j} is a monotone decreasing sequence (since as j becomes large we are taking the supremum of a smaller set of numbers), so it has a limit.  We define the limit superior of  {a_j} to be 

                                            lim sup a_j   = lim B_j      as j -> infinity

Remark      What is the intuitive concept of this definition?  For each j, A_j picks out the greatest lower bound of the sequence in the j^th  position or later. So the sequence {A_j} should tend to the smallest possible limit of any subsequence of  {a_j}.

    Likewise, for each j, B_j  picks out the least upper bound of the sequence  in the j^th  position or later.  So the sequence  {A_j} should tend to the greatest possible limit of any subsequence of {a_j}.   We shall make this remark more precise in Proposition 1 below.

    Notice that it is implicit in the definition that every real sequence has a limit supremum and a limit infimum.

Example 

Consider the sequence {(-1)^j}. Of course this sequence does not converge. Let us calculate its lim sup and lim inf.

    Referring to the definition, we have that A_j = -1 for every j.  So

                               lim inf (-1)^j  = lim (-1) = -1.

Similarly,  B_j  = +1 for every j. Therefore

                                lim sup (-1)^j  = lim (+1) = +1.

    As we predicted in the remark, the lim inf is the least subsequential limit, and the lim sup is the greatest subsequential limit.

Now let us prove the characterizing property of lim sup and lim inf to which we have been alluding.

PROPOSITION

Let  {a_j} be a sequence of real numbers.  Let B = lim sup a_j  as j -> infinity and  A  = lim inf a_j  as j -> infinity.  If  {a_jk } is any subsequence then 

                                                                    A < lim inf a_jk < lim sup a_jk < B     as k -> infinity

Moreover, there is a subsequence {a_jl} such that

                                                                    lim a_jl = A 

and another sequence {a_jm} such that 

                                                                    lim a_jm = B             as m -> infinity

PROOF     

For simplicity in this proof we assume that all lim sup's and lim inf's are finite.  

    We begin by considering the lim inf.  We adopt the notation of Definition 1.  There is a j₁ > 1 such that  |A₁ - a_j1| < 2 ^-1  .  We choose j₁ to be as small as possible.  Next we choose j₂ , necessarily greater than or equal to j₁ , such that j₂ is as small as possible and |a_j2 - A₂ | < 2 ^-2.  Continuing in this fashion, we select a_jk > a_{j k-1} such that |a_jk - A| < 2 ^-k-1 , and so forth.

    Recall that A_k  ->  A  = lim inf  a_j  as j -> infinity.  Now fix E  > 0.  If N is an integer so large that k > N implies that |A_k - A | < E /2 and also that 

2 ^{-N <}  E /2 , then for such k we have 

                            |a_jk  - A | <   | a_jk  - A_k | + | A_k  - A |

                                          < 2 ^-k  + E /2

                                          < E /2 + E /2

                                          = E  

Thus the subsequence {a_jk} converges to A , the lim inf of the given sequence.  A similar construction gives a (different) subsequence converging to B, the lim sup of the given sequence.

    Now let {a_jl} be any subsequence of the sequence {a_j}. Let B ^*  be the lim sup of this subsequence.  Then, by the first part of the proof, there is a subsequence {a_jlm} such that 

                                            lim a_jlm  =  B ^*       as m -> infinity

^But   a_jlm  < B_jlm   by the very definition of the B' s.  Thus 

                                            B ^* = lim a_jlm  <  B_jlm = B     as    m -> infinity.

or

                                                    lim sup a_jl  <  B            as    l -> infinity,

as claimed.  A similar argument shows that

                                                    lim inf  a_jl  > A                 as l  -> infinity.

This completes the proof of the proposition.

COROLLARY

If  {a_j}is a sequence  and  {a_jk} is a convergent subsequence then 

                                                   lim inf a_j < lim a_jk < lim sup a_j    as    j  -> infinity and as  as  k -> infinity.

Take it for granted for the moment that the number pi has been rigorously defined and proved to be irrational .  Then it can be shown that the positive integers are dense , modulo multiples of pi, in the interval [0, pi].  It follows that the sequence {cos j} is dense in the interval [-1,1] in the following sense: given any number A an element of  [-1,1] there is a subsequence cos j_k such that lim cos j_k = A as  k -> infinity.  In particular, the lim sup of the sequence  is 1 and the lim inf is -1.  

    I close with a fact that is similar to one you have seen before for the supremum and infimum.  This is given without proof.

PROPOSITION 2

Let {a_j} be a sequence and set lim sup a_j = B and lim inf a_j  = A.. Assume that  A , B are finite real numbers. Let E  > 0.  Then there are arbitrarily large j such that a_j > B  -  E  .  Also there are arbitrarily large k such that a_k < A  +  E   .

For more detailed information on lim sups and lim infs log onto 

http://www.shu.edu/html/teaching/math/reals/numseq/limsup.html