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Version 8 |
| Let $S\subset\reals$ be a set of real numbers. Recall that a limit |
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 |
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 |
$\epsilon>0$ there exist infinitely many $y\in S$ such that |
| $$\vert x-y\vert <\epsilon.$$ |
$$\vert x-y\vert <\epsilon.$$ |
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We define $\limsup S=\overline{\lim}$, pronounced the
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We define $\limsup S$, pronounced the
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| {\em limit superior} of $S$, to be the supremum of all the limit |
{\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 |
points of $S$. If there are no limit points, we define the limit |
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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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| We can generalize the above definition to the case of a |
We can generalize the above definition to the case of a |
| mapping $f:X\to\reals$. Now, we define a limit point of |
mapping $f:X\to\reals$. Now, we define a limit point of |
| $f$ to be an $x\in \reals$ such that for all |
$f$ to be an $x\in \reals$ such that for all |
| $\epsilon>0$ there exist infinitely many $y\in X$ such that |
$\epsilon>0$ there exist infinitely many $y\in X$ such that |
| $$\vert x-f(y)\vert <\epsilon.$$ |
$$\vert x-f(y)\vert <\epsilon.$$ |
| We then define $\limsup f$, to be the |
We then define $\limsup f$, to be the |
| supremum of all the limit points of $f$, or $-\infty$ if there are no |
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 |
limit points. |
| considering the limit superior of the inclusion mapping $\iota: S\to |
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| \reals$. |
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| Since a sequence of real numbers $x_0, x_1, x_2, ,\ldots$ is just a |
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 |
mapping from $\natnums$ to $\reals$, we may adapt the above definition |
| to arrive at the notion of the limit superior of a sequence. However |
to arrive at the notion of the limit superior of a sequence. However |
| for the case of sequences, an alternative, but equivalent definition |
for the case of sequences, an alternative, but equivalent definition |
| is available. For each $k\in\natnums$, let $y_k$ be the supremum of |
is available. For each $k\in\natnums$, let $y_k$ be the supremum of |
| the $k\supth$ tail, |
the $k\supth$ tail, |
| $$y_k = \sup_{j\geq k} x_j .$$ |
$$y_k = \sup_{j\geq k} x_j .$$ |
| This construction produces a |
This construction produces a |
| non-increasing sequence |
non-increasing sequence |
| $$y_0 \geq y_1 \geq y_2 \geq \ldots,$$ |
$$y_0 \geq y_1 \geq y_2 \geq \ldots,$$ |
| which either converges to its infimum, or diverges to $-\infty$. |
which either converges to its infimum, or diverges to $-\infty$. |
| We define the limit superior of the original sequence to be this limit; |
We define the limit superior of the original sequence to be this limit; |
| $$\limsup_{k} x_k = \lim_k y_k.$$ |
$$\limsup_{k} x_k = \lim_k y_k.$$ |
