Let the least term of a sequence be a term which is smaller than all but a finite number of the terms which are equal to . Then is called the lower limit of the sequence.

A lower limit of a series

is said to exist if, for every , for infinitely many values of and if no number less than has this property.

More things to try:

Weisstein, Eric W. "Lower Limit." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LowerLimit.html