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From Probability Through Problems By Marek Capinski,Tomasz Jerzy Zastawnaik

Verify that

$P(\lim_{n \to \infty}\inf A_n) \leq \lim_{n \to \infty}\inf P(A_n) \leq \lim_{n \to \infty}\sup P(A_n) \leq P(\lim_{n \to \infty}\sup A_n)$

Solution as given : Consider

$B_n=\cap_{k=n}^{\infty}A_k$

then

\begin{eqnarray*}P(\lim_{n \to \infty}\inf A_n) &=& P(\cup_{n=1}^{\infty}B_n)...since \space \lim_{n \to \infty}\inf A_n =\cup_{n=1}^{\infty}\cap_{k=n}^{\infty}A_k \\ &=& \lim_{n \to \infty}P(B_n)...since \space B_1\subset B_2\subset...\\ &=& \lim_{n \to \infty} \inf P(B_n) ..since \space P(B_1)\leq P(B_2) \leq...\\ &\leq& \lim_{n \to \infty}\inf P(A_n)...since\space B_n\subset A_n \\ \end{eqnarray*}

What I am not getting is the $3^{rd}$ step,I mean how $\lim_{n \to \infty}P(B_n)=\lim_{n \to \infty} \inf P(B_n)$?Please explain..

Thanks in advance..

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    $\begingroup$ Please avoid math-only titles. $\endgroup$
    – Asaf Karagila
    Mar 30 '20 at 16:38
  • $\begingroup$ Is $lim_{n\to\infty}inf$ suppose to be $\liminf_{n\to\infty}$? (Also \lim and \inf and \liminf will make it look better, same holds for \sup and \limsup.) $\endgroup$
    – Asaf Karagila
    Mar 30 '20 at 16:39
  • $\begingroup$ By the way, this is a special case of Fatou's lemma. If a sequence of real numbers $(a_n)_{n\in\mathbb N}$ converges, then $$\lim_{n\to\infty}a_n=\limsup_{n\to\infty}a_n=\liminf_{n\to\infty} a_n$$ $\endgroup$ Mar 31 '20 at 8:29
  • $\begingroup$ But why will the sequence {$P(B_n)$} will be converging?It will be converging in the case if there exist a $k\in N$ such that $B_k=B_{k+i}$ for all $i \in \mathbb{N}$ .and this is possible if $A_k=A_{k+i}$ for all $i \in \mathbb{N}$.But how can this be possible if the sample space is infinite $\endgroup$ Apr 1 '20 at 10:48
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I think there may be some mistake in the book.We can write proof as

$B_n=\cap_{k=n}^{\infty}A_k$

for all $k \geq n$,$B_n \subset A_k$

so we can say $P(B_n) \leq P(A_k)$ for all $k \geq n$

So,we can say $P(B_n) \leq \inf_{k \geq n} P(A_k)$....(1)

So we proceed as

\begin{eqnarray*}P(\lim_{n \to \infty}\inf A_n) &=& P(\cup_{n=1}^{\infty}B_n)...since \space \lim_{n \to \infty}\inf A_n =\cup_{n=1}^{\infty}\cap_{k=n}^{\infty}A_k \\ &=& \lim_{n \to \infty}P(B_n)...since \space B_1\subset B_2\subset...\\ &\leq & \lim_{n \to \infty} \inf_{k \geq n} P(A_k) ..from (1) \\ &=& \lim_{n \to \infty}\inf P(A_n) \\ \end{eqnarray*}

is it correct way?

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