Definition 6

The limit of the real-valued function $M \to \sup_{x \in M} f(x)$ with respect to the directed set $\mathfrak{G}$ is called the upper limit of $f$ with respect to the filter $\mathfrak{G}$, and is denoted by $\lim \sup_{\mathfrak{G}} f$, or by $\lim \sup_{x, \mathfrak{G}} f(x)$.

The lower limit of $f$ with respect to the filter $\mathfrak{G}$ is defined similarly, and is denoted by $\lim \inf_{\mathfrak{G}} f$ or $\lim \inf_{x, \mathfrak{G}} f(x)$. Thus we have

(10)

Often the filter $\mathfrak{G}$ is suppressed from the notation, and we write simply $\lim \sup f$ or $\lim \sup_{x} f(x)$, or $\lim \sup f(x)$ when there is no risk of confusion.

From formulae (10) and Theorem 1 we have

(11)

By Theorem 2 of no. 2 we may also write

(12)

Also we may replace the filter $\mathfrak{G}$, on the right-hand sides of formulae (10) and (12), by any base $\mathfrak{B}$ of $\mathfrak{G}$.

From (2) and (10),

and therefore we need consider only the upper limit.