(analysis) The lower limit of a sequence of real numbers is the real number which can be found as follows: remove the first term of the sequence in order to obtain the "first subsequence." Then remove the first term of the first subsequence in order to obtain the "second subsequence." Repeat the removal of first terms in order to obtain a "third subsequence," "fourth subsequence," etc. Find the infimum
of each of these subsequences, then find the supremum
of all of these infimums. This supremum is the lower limit.