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 an, the upper limit of an is defined to be
lim an = lim sup an =  lim  sup {an : 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.

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