3. Sequences of Numbers
3.4. Lim Sup and Lim Inf
When dealing with sequences there are two choices:
 the sequence converges
 the sequence diverges
Definition 3.4.1: Lim Sup and Lim Inf  
Let be a sequence of real
numbers. Define
A_{j} = inf{a_{j} , a_{j + 1} , a_{j + 2} , ...}and let c = lim (A_{j}). Then c is called the limit inferior of the sequence . Let be a sequence of real numbers. Define B_{j} = sup{a_{j} , a_{j + 1} , a_{j + 2} , ...}and let c = lim (B_{j}). Then c is called the limit superior of the sequence . In short, we have:

Proposition 3.4.3: Lim inf and Lim sup exist  
lim sup and lim inf always exist (possibly infinite) for any sequence of real numbers. 
It is important to try to develop a more intuitive understanding about lim sup and lim inf. The next results will attempt to make these concepts somewhat more clear.
Proposition 3.4.4: Characterizing lim sup and lim inf  
Let be an arbitrary
sequence and let
c = lim sup(a_{j}) and
d = lim inf(a_{j}).
Then

A little bit more colloquial, we could say:
 A_{j} picks out the greatest lower bound for the truncated sequences {a_{j}}. Therefore A_{j} tends to the smallest possible limit of any convergent subsequence.
 Similarly, B_{j} picks the smallest upper bound of the truncated sequences, and hence tends to the greatest possible limit of any convergent subsequence.
The final statement relates lim sup and lim inf with our usual concept of limit.
Proposition 3.4.6: Lim sup, lim inf, and limit  
If a sequence {a_{j}} converges then
lim sup a_{j} = lim inf a_{j} = lim a_{j}Conversely, if lim sup a_{j} = lim inf a_{j} are both finite then {a_{j}} converges. 
To see that even simple concepts like lim inf and lim sup can result in interesting math consider the following unproven conjecture:
If p_{n} is the nth prime number, then lim inf p_{n+1}  p_{n} = 2 and lim sup p_{n+1}  p_{n} =
The first equation is a conjecture, not yet proven, called the twin prime conjecture. In fact, it is not even known if the lim inf is finite. On the other hand, the second equation involving lim sup is known to be infinite because of arbitrary spaces between two primes.