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 an = lim sup an = lim sup {an : 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 an = lim inf an, then lim an exists.
- If lim an exists, then lim an = lim sup an.