limit superiorLimitSuperior2002-02-18 09:54:262005-03-28 12:07:56Definition\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\natnums}{\mathbb{N}}
\newcommand{\cnums}{\mathbb{C}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
\newcommand{\rb}{\right]}
\newcommand{\supth}{^{\text{th}}}
\newtheorem{proposition}{Proposition}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
{\em 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;
$$\limsup_{k} x_k = \lim_k y_k.$$