Lemmas about liminf and limsup in an order topology. #
If the liminf and the limsup of a filter coincide, then this filter converges to their common value, at least if the filter is eventually bounded above and below.
If a function has a limit, then its limsup coincides with its limit.
If a function has a limit, then its liminf coincides with its limit.
If the liminf and the limsup of a function coincide, then the limit of the function exists and has the same value
If a number
a is less than or equal to the
liminf of a function
f at some filter
and is greater than or equal to the
f tends to
a along this filter.
Assume that, for any
a < b, a sequence can not be infinitely many times below
b. If it is also ultimately bounded above and below, then it has to converge. This even
b are restricted to a dense subset.