Proposition 11

Let $f$ and $g$ be two real-valued functions defined on a filtered set $X$. Then the relation $f \leq g$ implies

This is an immediate consequence of the relations (12).

When $X$ is a topological space and $\mathfrak{G}$ is the neighbourhood filter of a point $a$ of $X$, we write $\limsup_{x \to a} f(x)$ [resp. $\liminf_{x \to a} f(x)$] in place of $\limsup_{\mathfrak{G}} f$ [resp. $\liminf_{\mathfrak{G}} f$]; clearly we have

More generally, if $X$ is a subspace of a topological space $Y$, and if $\mathfrak{G}$ is the trace on $X$ of the neighbourhood filter of a point $a \in \overline{X}$, we write $\limsup_{x \to a, x \in X} f(x)$ [resp. $\liminf_{x \to a, x \in X} f(x)$] instead of $\limsup_{\mathfrak{G}} f$ [resp. $\liminf_{\mathfrak{G}} f$] ; $\limsup$ is called the upper limit of $f(x)$ as $x$ tends to $a$ while remaining in $X$. If $X$ is the complement of $\{a\}$ we write "x \neq a" in place of "x \in X" in these notations.

If $A$ is a subset of $X$ such that $a \in \overline{A}$, then (Corollary 2 to Theorem 3)

If $V$ is a neighbourhood of $a$ in $Y$, we have (Corollary 3 to Theorem 3)

Hence the notions of upper and lower limits at a point of a topological space are, like the notion of limit, of local character.

Finally, if $\mathfrak{G}$ is the Fréchet filter on $\mathbf{N}$, the upper (resp. lower) limit, with respect to $\mathfrak{G}$, of the mapping $n \to u_n$ of $\mathbf{N}$ into $\overline{\mathbf{R}}$ is denoted by $\limsup_{n \to \infty} u_n$ (resp. $\liminf_{n \to \infty} u_n$) and is called the upper (resp. lower) limit of the sequence of real numbers $u_n$.

The relation $\limsup_{n \to \infty} u_n = a \in \mathbf{R}$ is therefore equivalent to the following : given any $\varepsilon > 0$ there exists an integer $n_0$ such that, for each $n \geq n_0$ we have $u_n \leq a + \varepsilon$, and for an infinity of values of $n$ we have $u_n \geq a - \varepsilon$. The definition of the upper limit of a sequence may be translated similarly when its value is $+\infty$ or $-\infty$.

Given a sequence $(f_n)$ of real-valued functions defined on a set $X$, we denote by $\limsup_{n \to \infty} f_n$ (resp. $\liminf_{n \to \infty} f_n$) the real-valued function whose value at any point $x \in X$ is $\limsup_{n \to \infty} f_n(x)$ [resp. $\liminf_{n \to \infty} f_n(x)$]. From (10) and (12) we deduce

the limits being taken in the product space $\overline{\mathbf{R}}^X$. The sequence $(f_n)$ has a limit in $\overline{\mathbf{R}}^X$ if and only if $\limsup_{n \to \infty} f_n = \liminf_{n \to \infty} f_n$ (Corollary 1 to Theorem 3, and Chapter I, § 7, no. 6, Corollary 1 to Proposition 10).