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Increasing and decreasing sequence of events

Would any member of Cross validated stack exchange prove highlighted 1 and 2 by simple and clear explanation with examples?

My answers:

  1. A sequence of events {$E_n, n\geq 1$} is said to be an increasing sequence if $E_n \subset E_{n+1}, n\geq 1 $ That means $E_n$ consists of those points that are not in any of the earlier $E_i, i < n$ Hence $E_{n+1},E_n, E_{n-1},...,E_1$ are matually exclusive events. So, $$\lim_{n \to \infty} E_n = \cup^{\infty}_{i=1} E_i$$

  2. If {$E_n, n\geq 1$} is a decreasing sequence, the {$E^c_n, n \geq 1 $} is an increasing sequence. Hence $$\cup^{\infty}_{1} E^c_n = \lim_{n\to \infty} E^c_n$$

But, as $\cup^{\infty}_1 E^c_n = (\cap^{\infty}_1 E_n)^c$ , we see that $$1-\cap^{\infty}_1 E_n = \lim_{n \to \infty} [ 1- E_n]$$

or equivalently,$$\cap^{\infty}_1 E_n = \lim_{n \to \infty} E_n$$

Dhamnekar Winod
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  • Please type your question as text, do not just post a photograph or screenshot (see [here](https://stats.meta.stackexchange.com/a/3176/)). When you retype the question, add the [tag:self-study] tag & read [its wiki](https://stats.stackexchange.com/tags/self-study/info). Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. – kjetil b halvorsen Feb 21 '22 at 13:33
  • There is nothing to "prove:" these are definitions. It looks like you seek actual examples. Two situations involving detailed, worked calculations of this sort appear here at https://stats.stackexchange.com/a/164995/919 and https://stats.stackexchange.com/a/221555/919. – whuber Feb 21 '22 at 13:33

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