Let $S\subset\reals$ be a set of real numbers. Recall that a limit point of $S$ is a real number $x\in\reals$ such that for all $\epsilon>0$ there exist infinitely many $y\in S$ such that $$\vert x-y\vert <\epsilon.$$ We define $\liminf S$ , pronounced the limit inferior of $S$ , to be the infimum of all the limit points of $S$ . If there are no limit points, we define the limit inferior to be $+\infty$ .

The two most common notations for the limit inferior are $$\liminf S$$ and $$\underline{\lim}\, S\,.$$

An alternative, but equivalent, definition is available in the case of an infinite sequence of real numbers $x_0, x_1, x_2, ,\ldots$ . For each $k\in\natnums$ , let $y_k$ be the infimum of the $k\supth$ tail, $$y_k = \inf_{j\geq k} x_j .$$ This construction produces a non-decreasing sequence $$y_0 \leq y_1 \leq y_2 \leq \ldots,$$ which either converges to its supremum, or diverges to $+\infty$ . We define the limit inferior of the original sequence to be this limit; $$\liminf_{k} x_k = \lim_k y_k.$$