Currently I am taking a lecture which is based on the book 'Probability with Martingales' by David Williams. So far we are on the topic of infinite product measures and I'm struggling with the proof of the following lemma which belongs to the existence of such an infinite product measure:
Let $\Omega $ be the set of all real-valued sequences and $\mathcal{A}_k $ the $\sigma$-Algebra generated by the sets $\{(\omega_n):\omega_1 \in B_1,\dots,\omega_k \in B_k\} $, with $B_1,\dots,B_k\in \mathcal{B}(\mathbb{R})$. Elements of $\mathcal{A_k}$ can be written as $$ A_k=\{(\omega_n):(\omega_1,\dots,\omega_k)\in B_k\} \text{ for } B_k \in \mathcal{B}(\mathbb{R^k}).$$ Let $\mathcal{A}_0$ be the algebra defined as $$\mathcal{A}_0 := \bigcup_{1 \leq \ell \leq k} \mathcal{A}_\ell$$ and define $P_0:\mathcal{A}_0 \rightarrow [0,1] $ as $$ P_0(\{(\omega_n):(\omega_1,\dots,\omega_k) \in B_k \})=P_1 (B_1)\otimes\dots\otimes P_k(B_k).$$
Show: If $(H_n)$ is a falling sequence of sets in $\mathcal{A}_0$ and $\epsilon>0$ exists so that $P_0(H_n)\geq \epsilon$ for all $n$, then $\cap_n H_n \neq \emptyset.$
Although I have seen that a proof is given on p.229 in 'Probability with Martingales', I am still confused because it includes topics my professor did not talk about yet and also because the whole notation of it is overwhelming me a bit.