# Show almost sure convergence of an intersection

I am working on the following exercise:

Let $$(X_n)_{n \ge 1}$$ be a stochastic process of i.i.d. random variables with values in a finite set $$\mathcal{X}$$ and distribution $$p(x)$$. Let $$\mu= E(X_1)$$. Given $$\epsilon > 0$$, let

\begin{align} A^{(n)} &:= \biggl\{ (x_1, \ldots, x_n) \in \mathcal{X} : \bigg\lvert -\frac{1}{n} \log_2\bigl((p(x_1,\ldots,x_n) \bigr) - H(X_1) \bigg\rvert \le \epsilon \biggr\} \\ B^{(n)} &:= \biggl\{ (x_1, \ldots, x_n) \in \mathcal{X} : \bigg\lvert -\frac{1}{n} \sum_{i=1}^nx_i-\mu \bigg\rvert \le \epsilon \biggr\} \end{align}

Show that

a) $$P \left[\bigl(X_1,\ldots,X_n\bigr) \in A^{(n)}\right] \rightarrow1 \text{ for } n \rightarrow \infty$$

b) $$P \left[\bigl(X_1,\ldots,X_n\bigr) \in B^{(n)}\right] \rightarrow1 \text{ for } n \rightarrow \infty$$

c) $$P \left[\bigl(X_1,\ldots,X_n\bigr) \in A^{(n)} \cap B^{(n)}\right] \rightarrow1 \text{ for } n \rightarrow \infty$$

I have done $$a)$$ and $$b$$). I do not see how I could do c) however. Could you help me?

To save some writing, let $$E_n$$ be the event that $$(X_1, \dots X_n) \in A^{(n)}$$ and $$F_n$$ similarly for $$B^{(n)}$$. You have shown $$P(E_n) \to 1$$ and $$P(F_n) \to 1$$, and you want to show $$P(E_n \cap F_n) \to 1$$.
Use de Morgan's law to write $$P(E_n \cap F_n) = 1 - P(E_n^c \cup F_n^c)$$. Then by union bound, $$P(E_n^c \cup F_n^c) \le P(E_n^c) + P(F_n^c) \to 0.$$