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## Homework Statement

Let A1,A 2,...be an inﬁnite sequence of events such that A1⊂A2⊂.... Prove that

Pr(∪A

_{i})=lim

_{n→inf}Pr(A

_{n})

∪ is also an iterator that starts from i=1 to infinity. How can you put those iterators?[/B]

## Homework Equations

I decided to use induction

## The Attempt at a Solution

Proof by induction

Base case

let A

_{1}⊂A

**[/B]**

_{2}so

**A**

then we have to show that if A

so

This is true because

so by math induction

_{1}∪A_{2}=A

Therefore Pr(_{2}Therefore Pr(

**A**_{1}∪A_{2})=Pr(A

Inductive Step

let_{2})Inductive Step

let

**Pr(∪A**_{i})=lim_{k→inf}Pr(A_{k})then we have to show that if A

_{k}⊂A_{k+1}then lim_{k→inf}Pr(A_{k}∪A_{k+1})=lim_{k→inf}Pr(A_{k+1})so

This is true because

**A**_{k}∪A_{k+1}=A

so_{k+1}so

**lim**_{k→inf}Pr(A_{k}∪A_{k+1})=lim_{k→inf}Pr(A_{k+1})so by math induction

**Pr(∪A**

_{i})=lim_{n→inf}Pr(A_{n})

is that right?is that right?