Proof that $\frac{S_n}{n}$ converges almost surely to $\mu$ I'm trying to show that given $(X_i)$ i.i.d., $E[X_i^2] < \infty$, $\mu := E[X_i]$ then $P\Big [  \lim_{n \rightarrow \infty} \frac{S_n}{n} = \mu \Big ] = 1$ where $ S_n := \sum_{k=1}^n X_k$.
So far, I have rewritten  $P\Big [  \lim_{n \rightarrow \infty} \frac{S_n}{n} = \mu \Big ]$ as 
$$ \lim_{k \rightarrow \infty} P \Big [ \cap_{n \geq k} \{ \omega \Big | |\frac{S_n}{n} - \mu| < \varepsilon \}  \Big ]$$
But I'm not sure how to proceed from here. I have
$$ \sum_{n} P\Big [ |X_n - X|  > \varepsilon \Big] < \infty  \Rightarrow P\Big [ \lim_n X_n = X \Big ] = 1$$ which I think I should apply but I don't see how. Can anyone help me with this? Also, I don't see where $E[X_i^2] < \infty$ comes in. Many thanks for your help!
 A: A proof of the strong law of large numbers can be more
or less complicated depending on your hypotheses.
In your case, since you assume that $E[X_i^2]<\infty$ there is
a straightforward proof. I am taking this from section 7.4 of the third
edition of Probability and Stochastic Processes by Grimmett and Stirzaker.
First, by splitting into positive and negative parts we can assume (without
loss of generality) that $X_i\geq 0$.
Second, using the positivity, it suffices to
prove that  $S_{n^2}/n^2\to\mu$ almost surely; that is, we only need
convergence along that subsequence.
Next, Chebyshev's inequality gives
$$P(|S_{n^2}/n^2-\mu|>\varepsilon_n)\leq{E[X_i^2]\over n^2\varepsilon_n^2}.$$
Choosing $\varepsilon_n\downarrow 0$ so slowly that the right hand side above
is summable, Borel-Cantelli finishes the job since then
$$P(|S_{n^2}/n^2-\mu| \leq \varepsilon_n \mbox{ for all but finitely many }n) = 1.$$
In fact, the strong law of large numbers holds under the weaker hypothesis
that $E[|X_i|]<\infty$. There are various proofs in the literature, but every
student of probability ought to be familiar with Etemadi's tour de force elementary
proof. Etemadi uses a clever truncation argument and similar tools to those  above, and only needs pairwise independence of the $X_i$'s, not full independence.
Some good textbooks like Grimmett and Stirzaker (section 7.5), Billingsley's Probability and Measure (2nd edition), or Durrett's Probability: Theory and Examples (2nd edition) include Etemadi's treatment.
N. Etemadi, An elementary proof of the strong law of large numbers,
Z. Wahrscheinlichkeitstheorie verw. Gebeite 55, 119-122 (1981)
