I am trying to understand ideas like Almost sure convergence and convergence of sequences of random variables.
I am trying to prove:
Given an infinite sequence of random variables $X_i,i\in\mathbb{N}$ on some probability space ($\Omega$,F,P) which are iid with $\mathbb{E}(X_i)=\mu<\infty,Var(X_i)=\sigma^2<\infty$ and $S_n=\sum_{i=1}^nX_i$. Prove $\frac{1}{n^p}S_{n^p}\overset{a.s}\to\mu$ for some natural number p.
In order to prove this, first I define almost sure convergence by creating event $E$ where
$$E=\left\{ \omega\in\Omega:\lim_{n\to\infty}\frac{1}{n^p}S_{n^p}(\omega)=\mu\right\}.$$ This is the set of words $\omega$ such that plugging them into the function $\frac{1}{n^p}S_{n^p}(\omega)$ goes to $\mu$ as $n\to\infty$.
I then wanted to define what it meant for a random variable to limit to some $\mu$ so i created set $A_n$ such that $$A_n(l)=\left\{\omega\in\Omega:\frac{1}{n^p}S_{n^p}(\omega)\in(\mu-l,\mu+l)\right\}.$$
This is the set of $\omega$ that for a given random variable sit in a fixed radius of at most $l$, and an idea would be to make $l$ arbitrarily small as such:
$$E=\bigcap_{l=1}^\infty\bigcup_{N=1}^\infty\bigcap_{n=N+1}^\infty A_n \left(\frac{1}{l} \right).$$
Here this is the event where for all $\omega$ such that given any $l$ if there is some $N$ such that for all $n>N$ then $\frac{1}{n^p}S_{n^p}(\omega)\in(\mu-l,\mu+l)$ (This bit is confusing and im not entirely sure this is right).
and through DeMorgan's Law $$E^c=\bigcup_{l=1}^\infty\bigcap_{N=1}^\infty\bigcup_{n=N+1}^\infty A_n^c \left(\frac{1}{l}\right)$$
and then we can use the inequality
$$\mathbb{P}\left(\bigcup_{l=1}^\infty\bigcap_{N=1}^\infty\bigcup_{n=N+1}^\infty A_n^c\left(\frac{1}{l}\right)\right)\leq\sum_{k=1}^{\infty}\mathbb{P}\left(\bigcap_{N=1}^\infty\bigcup_{n=N+1}^\infty A_n^c \left(\frac{1}{l}\right)\right).$$
My questions are:
Is the structure of this proof correct and what can I improve?
How do I prove that my event has measure 1 (Proving absolute convergence)
Does this logic generalise to some arbitrary element?
Any advice or suggested reading material would be greatly appreciated, thank you.