How would you go about doing this? is it really obvious?
So where are you having trouble? Is it in:
The first part : Reading set notations(about the infinitely often)
The second part : The union bound axiom? the one that says probability of union of countable number of events is upperbounded by the sum of probability of those events
In case of basic application problems like these, its beneficial to both (us and you), if you tell us where you are having trouble.
Hello,
If $\displaystyle B_n$ happens, it means that there is at least one $\displaystyle A_m$ in its union that happens.
This can be translated as "$\displaystyle \exists m\geq n$, such that $\displaystyle A_m$ happens"
Then, for $\displaystyle A=\bigcap_{n\geq 1} B_n$ to happen, $\displaystyle B_n$ has to happen for any $\displaystyle n\geq 1$
This can finally be translated as "for any n, there exists m>n such that $\displaystyle A_m$ happens".
In other words, there are infinitely many $\displaystyle A_m$ events that happen.
You can read this for more things : http://www.mathhelpforum.com/math-he...up-liminf.html in particular post #4, by Laurent
Probability axioms - Wikipedia, the free encyclopediaand also showing how the P(A)≤P(Bn). is it because A is a subset of Bn?
Since A is, as you noticed, a subset of $\displaystyle B_n$, then there exists a set $\displaystyle E$, disjoint from A, such that $\displaystyle A\cup E=B_n$
By the third axiom, we have (the red equality) :
$\displaystyle \mathbb{P}(B_n)=\mathbb{P}(A\cup E){\color{red}=}\mathbb{P}(A)+\mathbb{P}(E)$
Now, by the first axiom, we know that $\displaystyle \mathbb{P}(E)\geq 0$
And it follows that $\displaystyle \mathbb{P}(B_n)\geq \mathbb{P}(A)$
Note that P($\displaystyle A_n$ happens infinitely often)=0 $\displaystyle \Leftrightarrow$ P(A)=0and how does this relate to the third part?
And note that if $\displaystyle \sum_{m\geq 1} a_m<\infty$, this implies that the partial reminder (I don't know the exact name) $\displaystyle \sum_{m\geq n} a_m \xrightarrow[]{n\to\infty} 0$
It should be okay from now
Hmm this axiom isn't stated in the probability axioms. Nor it is stated in the axioms defining a measure. Though it's a direct consequence. I know the French name, but not sure in English... I think it is called sigma subadditivity or something like that...The second part : The union bound axiom? the one that says probability of union of countable number of events is upperbounded by the sum of probability of those events