# limit superior

Let $S\subset\mathbb{R}$ be a set of real numbers. Recall that a limit point of $S$ is a real number $x\in\mathbb{R}$ such that for all $\epsilon>0$ there exist infinitely many $y\in S$ such that

 $|x-y|<\epsilon.$

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 $f:X\to\mathbb{R}$. Now, we define a limit point of $f$ to be an $x\in\mathbb{R}$ such that for all $\epsilon>0$ there exist infinitely many $y\in X$ such that

 $|x-f(y)|<\epsilon.$

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 $\iota:S\to\mathbb{R}$.

Since a sequence of real numbers $x_{0},x_{1},x_{2},,\ldots$ is just a mapping from $\mathbb{N}$ to $\mathbb{R}$, 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 $k\in\mathbb{N}$, let $y_{k}$ be the supremum of the $k^{\text{th}}$ 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;

 $\limsup_{k}x_{k}=\lim_{k}y_{k}.$
Title limit superior LimitSuperior 2013-03-22 12:21:58 2013-03-22 12:21:58 rmilson (146) rmilson (146) 12 rmilson (146) Definition msc 26A03 limsup supremum limit LimitInferior