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.