Probability space proof 
PROBLEM
Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $(E_n)$ be a sequence in the $\sigma$-algebra $\mathcal{F}$.
$a)$ If the sequence $(E_n)$ is increasing (in the sence that $E_n \subset E_{n+1}$) with limit $E = \cup_nE_n$, prove that $P(E_n) \rightarrow P(E)$ as $n \rightarrow \infty$
$b)$ If the sequence $(E_n)$ is decreasing with limit $E$, prove that $P(E_n) \rightarrow P(E_n)$ as $n \rightarrow \infty$

MY APPROACH
$a)$ We know that $E_n \subset E_{n+1}$ so obviously $E_{n+1}$ consists of all sequences $E_i$ for $1 \leq i \leq n$.
$b)$ We know that the sequence is decreasing so $E_{n+1} \subset E_n$, so $E_1$ consists of all sequences $E_i$ for $ 2 \leq i \leq n$

I don't know how to formulate these proofs properly. I hope someone could help me..

a) Define $A_n:=E_{n}\setminus E_{n-1}$:  and $A_0:=E_0$. Then 


*

*if $i\neq j$, we have $A_i\cap A_j=\emptyset$;

*for each $N$, $\bigcup_{i=1}^NA_i=\bigcup_{i=1}^NE_i$;

*$\mathbb P\left(\bigcup_{i=1}^NA_i\right)=\sum_{i=1}^N\mathbb P(A_i)$.


b) Consider the sequence $(\Omega\setminus E_n)_n$. This forms a non-decreasing sequence, hence using the result of the first part: 
$$\mathbb P\left(\bigcup_{N=1}^\infty(\Omega\setminus E_N)\right)=\lim_{n\to\infty}\mathbb P\left(\bigcup_{j=1}^n(\Omega\setminus E_j)\right).$$
The LHS is $1-\mathbb P\left(\bigcap_{N=1}^\infty E_n\right)$ while the RHS is $1-\lim_{n\to \infty}\mathbb P\left(\bigcap_{j=1}^nE_j\right)$.
