The supremum limit, also called the limit superior or upper limit, of a sequence of real numbers 
, written 
and pronounced 'lim-soup,' is the limit
of
 |
as 
, where 
denotes the supremum.
Note that, by definition, 
is nonincreasing and so has a limit
in the extended real numbers, possibly
or 
. For example, suppose 
, then for 
odd, 
, and for 
even, 
. Another example is 
, in which case 
is a constant sequence 
.
When 
is finite, the sequence converges
to that real number
 |
If the two values differ, the sequence does not converge even in the extended real numbers.
If they agree at 
or 
,
it converges only in the extended-real sense.
