Given a sequence of real numbers 
, the supremum limit (also called the limit superior or upper
limit), written 
and pronounced 'lim-soup,' is the limit of
 |
as 
, where 
denotes the supremum.
Note that, by definition, 
is nonincreasing and so either has a limit or tends to 
. For example, suppose 
, then for 
odd, 
, and for 
even, 
. Another example is 
, in which case 
is a constant sequence 
.
When 
,
the sequence converges to the real number
 |
Otherwise, the sequence does not converge.
