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Finding the arrows very confusing in this context

i simply do not understand the arrows in this context! :\

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The two uparrows $\uparrow$ have different meanings.

$A_n \uparrow A$ means that $A_n$ is an increasing sequence of sets, i.e. $A_n \subseteq A_{n+1}$, and the countable union $\cup_{n \in \mathbb{N}} A_n = A$.

$P(A_n) \uparrow P(A)$ means that $P(A_n)$ is an increasing sequence of real numbers, i.e. $P(A_n) \leq P(A_{n+1})$. Since $P(E) \leq 1$, such a sequence converges to $L \in \mathbb{R}$ (why?). Part (a) asks you to show this limit $L$ is indeed $P(A)$.

The downarrows $\downarrow$ are just an analogue in the context of probability.

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$A_n \uparrow A$ means $A_m \subseteq A_n$ for $m \leq n$ and $A = \bigcup_{n \in \mathbb{N}}A_n$.

Similarly, $A_n \downarrow A$ means $A_m \supseteq A_n$ for $m \leq n$ and $A = \bigcap_{n \in \mathbb{N}} A_n$.

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    $\begingroup$ What about in the case $P(A_n)\uparrow P(A)$, when they are real numbers? $\endgroup$ – Regret Jan 2 '15 at 8:47
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    $\begingroup$ @regret $\mathbb{P}(A_m) \leq \mathbb{P}(A_n)$ for $m \leq n$ and $\mathbb{P}(A) = \sup_{n \in \mathbb{N}} \mathbb{P}(A_n)$. $\endgroup$ – saz Jan 2 '15 at 8:49

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