| Version 4 |
Version 3 |
| Let $S\subset\reals$ be a set of real numbers. Recall that a limit |
For $S\subset\reals$, let $S'$ be the set of limit points of $S$, and |
| point of $S$ is a real number $x\in\reals$ such that for all |
define |
| $\epsilon>0$ there exist infinitely many $y\in S$ such that |
$\limsup S,$ |
| $$\vert x-y\vert <\epsilon.$$ |
the {\em limit superior} of $S$, to be the supremum of |
| We define $\limsup S$, pronounced the |
$S'$. If $S'$ is empty, we say that the limit superior is $+\infty$. |
| {\em limit superior} of $S$, to be the supremum of all the limit |
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| points of $S$. If there are no limit points, we define the limit |
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| superior to be $+\infty$. |
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| The two most common notations for the limit superior are |
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| $$\limsup S$$ and |
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| $$\overline{\lim}\, S\,.$$ |
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| An alternative, but equivalent, definition is available in the case of |
An alternative, but equivalent, definition is available in the case of |
| an infinite sequence of real numbers $x_0, x_1, x_2, ,\ldots$. For |
an infinite sequence of real numbers $x_0, x_1, x_2, ,\ldots$. For |
| each $k\in\natnums$, let $y_k$ be the maximum of $x_0,\ldots, x_k$. |
each $k\in\natnums$, let $y_k$ be the maximum of $x_0,\ldots, x_k$. |
| This construction produces a non-decreasing sequence |
Note that |
| $$y_0 \leq y_1 \leq y_2 \leq \ldots$$ |
$$y_0 \leq y_1 \leq y_2 \leq \ldots$$ |
| We define the limit superior of the original sequence to be the |
We then define |
| limit of the maxima, |
$$\limsup_{k\rightarrow\infty} x_k = \lim_{k\rightarrow\infty} y_k.$$ |
| $$\limsup_{k\rightarrow\infty} x_k = \lim_{k\rightarrow\infty} |
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| y_k.$$ |
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