The symbol for a

limit of any

sequence of reals, known as the

*limit superior*,

*upper limit*, or the

*greater limit*.
For a real sequence a

_{n}, the upper limit of a

_{n} is defined to be

___
lim a_{n} = lim sup a_{n} = lim sup {a_{n} : n > N}
N→∞

(The lim symbol with a bar over it is another symbol for the upper limit.
The lim symbol with a bar under it is another symbol for the lower limit. See node

supremum for definition of sup.
)

The upper limit always exists for sequences of reals, and it can be either a real, -∞, or +∞.
Some results from

real analysis include:

- If lim sup a
_{n} = lim inf a_{n}, then lim a_{n} exists.
- If lim a
_{n} exists, then lim a_{n} = lim sup a_{n}.