nLab inferior limit

Inferior and superior limits

Idea

The inferior and superior limits ($\lim\,\inf$ and $\lim\,\sup$, or lower and upper limits) of an infinite sequence of real numbers are two generalizations of the usual limit that exist more often (indeed always, if one allows sufficiently general answers). This notion can be generalized far beyond its original context.

Terminology and notation

The inferior limit is denoted $\lim\,\inf$ or $\underline{\lim}$, while the superior limit is denoted $\lim\,\sup$ or $\overline{\operatorname{li̇m}}$. These symbols come from the Latin ‘limes inferior’ and ‘limes superior’ (with Latin plurals ‘limites inferiores’ and ‘limites superiores’); saying ‘limit inferior’ or ‘limit superior’ in English, while common, is like saying ‘logarithm natural’ because the symbol for the natural logarithm is ‘$\ln$’ (from the Latin ‘logarithmus naturalis’). Another variation is to read ‘$\lim\,\inf$’ as ‘limit infimum’ and ‘$lim\,\sup$’ as ‘limit supremum’, although this is etymologically incorrect. Sometimes one sees the more fully translated terms ‘lower limit’ and ‘upper limit’. On the other hand, in German, untranslated Latin is most common.

Definitions

Let $L$ be a complete lattice with a convergence structure, and let $F$ be a filter on (the underlying set of) $L$ (not a filter in $L$). Then the inferior limit of $F$ is the limit of the infima of the sets in $F$ (if this limit exists):

$$\lim\,\inf F = \lim_{A \in F} \inf A .$$

Similarly, the superior limit of $F$ is the limit of the suprema of the sets in $F$:

$$\lim\,\sup F = \lim_{A \in F} \sup A .$$

(Of course, if the convergence structure is not Hausdorff, then there may be multiple inferior and superior limits.) Here we are taking $F$ to define a couple of nets in $L$, each indexed by the sets in $F$; as is usual with filters, we take $A \leq B$ iff $B \subseteq A$. If $L$ is merely a poset and not a complete lattice, then we still use the same definitions, but they can only exist if $\inf A$ or $\sup A$ exists for sufficiently small $A \in F$.

If $F$ is merely a filter base, then precisely the same formulas give the inferior and superior limits of the filter generated by $F$. Of course, we can also start with anything else that generates a filter in some way, such as a sequence or more generally a net. In this case, we can write

$${\lim\,\inf}_n\, x_n = {\lim}_n\, \inf_{m \geq n} x_m$$

and

$${\lim\,\sup}_n\, x_n = {\lim}_n\, \sup_{m \geq n} x_m .$$

Given a set $X$, a filter $F$ on $X$, and a partial function $f$ to $L$ from $X$, the upper and lower limits of $f$ in the direction given by $F$ are the upper and lower limits of $f(F)$; that is,

$${\lim\,\inf}_F f = \lim_{A \in F} \inf f(A)$$

and

$${\lim\,\sup}_F f = \lim_{A \in F} \sup f(A) .$$

In particular, if $X$ is a pretopological space and $c$ is a adherent point of the domain of $f$ in $X$, then we can use the neighbourhood filter of $c$; similarly, if $c$ is an accumulation point of the domain of $f$ in $X$, then we can use the filter of punctured neighbourhoods of $c$. Then

$$\underset{x \to c}{\lim\,\inf}\, f(x) = \lim_{c \stackrel{\circ}\in A} \inf f(A)$$

and

$$\underset{x \to c}{\lim\,\sup}\, f(x) = \lim_{c \stackrel{\circ}\in A} \sup f(A)$$

if we use neighbourhoods (so that the limit must equal $f(c)$ if both exist, as traditionally taught in French), or

$$\underset{x \to c}{\lim\,\inf}\, f(x) = \lim_{c \stackrel{\circ}\in A} \inf f(A\setminus\{c\})$$

and

$$\underset{x \to c}{\lim\,\sup}\, f(x) = \lim_{c \stackrel{\circ}\in A} \sup f(A\setminus\{c\})$$

if we use punctured neighbourhoods (so that $f(c)$ is irrelevant to the limit, as traditionally taught in English). We can also use a neighbourhood base at $c$ instead of all neighbourhoods; in particular, if $X$ is a topological space with $O$ as its collection of open subsets, then we may use the (possibly punctured) open neighbourhoods of $c$; then

$$\underset{x \to c}{\lim\,\inf}\, f(x) = \lim_{c \in G \in O} \inf f(G)$$

and

$$\underset{x \to c}{\lim\,\sup}\, f(x) = \lim_{c \in G \in O} \sup f(G)$$

for the unpunctured limit, or

$$\underset{x \to c}{\lim\,\inf}\, f(x) = \lim_{c \in G \in O} \inf f(G\setminus\{c\})$$

and

$$\underset{x \to c}{\lim\,\sup}\, f(x) = \lim_{c \in G \in O} \sup f(G\setminus\{c\})$$

for the punctured limit.

Every partial order on a set defines a convergence structure (the order convergence?) under which a net $(x_n)_n$ converges to a point $x_\infty$ iff there exist a monotone increasing net $y$ and a monotone decreasing net $z$ such that $x_\infty = \sup_n y_n = \inf_n z_n$ and, for each $y$-index $i$ and $z$-index $j$, it is $n$-eventually true that $y_i \leq x_n \leq z_j$. In this case, we can also write

$$\lim\,\inf F = \sup_{A \in F}\, \inf A$$

and

$$\lim\,\sup F = \inf_{A \in F}\, \sup A .$$

The reason is that the net $(\inf A)_{A \in F}$ is monotone increasing, so that its limit in the order convergence is the same as its supremum (using itself for $y$ and a constant net for $z$); similarly, $(\sup A)_{A \in F}$ is monotone decreasing, with its limit the same as its infimum.