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 $\limsup S=\overline{\lim}$ , pronounced the limit superior of $S$ , to be the supremum of all the limit points of $S$ . If there are no limit points, we define the limit superior to be $-\infty$ .

We can generalize the above definition to the case of a mapping $f:X\to\reals$ . Now, we define a limit point of $f$ to be an $x\in \reals$ such that for all $\epsilon>0$ there exist infinitely many $y\in X$ such that $$\vert x-f(y)\vert <\epsilon.$$ We then define $\limsup f$ , to be the supremum of all the limit points of $f$ , or $-\infty$ if there are no limit points. We recover the previous definition as a special case by considering the limit superior of the inclusion mapping $\iota: S\to \reals$ .

Since a sequence of real numbers $x_0, x_1, x_2, ,\ldots$ is just a mapping from $\natnums$ to $\reals$ , we may adapt the above definition to arrive at the notion of the limit superior of a sequence. However for the case of sequences, an alternative, but equivalent definition is available. For each $k\in\natnums$ , let $y_k$ be the supremum of the $k\supth$ tail, $$y_k = \sup_{j\geq k} x_j .$$ This construction produces a non-increasing sequence $$y_0 \geq y_1 \geq y_2 \geq \ldots,$$ which either converges to its infimum, or diverges to $-\infty$ . We define the limit superior of the original sequence to be this limit; $$\limsup_{k} x_k = \lim_k y_k.$$