Let $X_n$ be a sequence of independent random variables, with $$\mathbf P(X_n=\pm(n+1))=\frac{1}{2(n+1)\ln(n+1)},\ \ \ \mathbf P (X_n=0)=1-\frac{1}{(n+1)\ln(n+1)}.$$
The task is to show that the Strong Law of Large Numbers does not apply here for $S_n=\sum_{k=1}^nX_k$.
To do that we define $A_n:=\{|X_n|\geq n\}=\{|X_n|=n+1\}$. Then $$\sum_{n\geq 1}P(A_n)=\sum_{n\geq 1}\frac{1}{(n+1)\ln(n+1)}=+\infty.$$ Therefore by Borel Cantelli Lemma $P(A_n, i.o.)=1$.
Then the argument is to use $A_n\subset\{|S_n|/n\geq1/2\}\cup\{|S_{n-1}|/n\geq1/2\}$. But this inclusion I do not get. Can somebody explain it?
I also would like to know why the curly brackets are not displayed correctly (i.o.w. why do i need to use \\} to display curly braces in preview??
Suppose $\frac{1}{n}\lvert S_{n-1} \rvert < \frac{1}{2}$ and $\lvert X_n \rvert \geq n$. Then $$ \frac{1}{n}\lvert S_n \rvert = \frac{1}{n}\lvert S_{n-1} + X_n \rvert \geq \frac{1}{n} \lvert X_n \rvert - \frac{1}{n} \lvert S_{n-1} \rvert \geq 1 - \frac{1}{2} = \frac{1}{2}.$$
Therefore if $A_n$ holds, we must either have that $\{ \frac{1}{n} S_{n-1} \geq \frac{1}{2} \}$ or $\{ \frac{1}{n} S_n \geq \frac{1}{2} \}$, which gives the required inclusion.