I'm currently working on some assignment and I'm not sure if my solution would be considered as correct in any way:
Let $(X)_{n \gt 1}$ be a stochastic process of i.i.d. random variables with values in a finite set $\mathcal{X}$ and distribution $p(x)$. Let $\mu = \mathbb{E} (X_1)$. Given $\varepsilon \gt 0$, show that
1.) $\mathbb{P}\left(\left|-\frac{1}{n} \log_2 p(x_1, \ldots, x_n) - H(X_1)\right| \leq \varepsilon\right) \rightarrow 1$
2.) $\mathbb{P}\left( \left| \frac{1}{n} \sum^{n}_{i=1} x_i - \mu \right| \leq \varepsilon \right) \rightarrow 1$
for $n \rightarrow \infty$.
If I rewrite 1. as
$\mathbb{P}\left(\left|-\frac{1}{n} \sum^{n}_{i = 1} \log_2(p(x_i)) - \mathbb{E}(-\log_2p(x_1))\right| \leq \varepsilon \right) \rightarrow1$
both 1. and 2. seem to be true due to the Weak Law of Large Numbers.
I have the feeling though that this is a little bit too easy and there must be something more to it.