lim sup (thing)

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

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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 a_n = lim inf a_n, then lim a_n exists.
• If lim a_n exists, then lim a_n = lim sup a_n.