Lower Limit

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

A lower limit of a Series

$$\mathop{\rm lower} \lim_{n\to \infty} S_n = \underline{\lim_{n\to \infty}} S_n =h$$

is said to exist if, for every $\epsilon>0$, $\vert S_n-h\vert < \epsilon$ for infinitely many values of $n$ and if no number less than $h$ has this property.

See also Limit, Upper Limit

References

Bromwich, T. J. I'a and MacRobert, T. M. ``Upper and Lower Limits of a Sequence.'' §5.1 in An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea, p. 40 1991.