Prove that $\epsilon$ is an elementary family of sets

Question

My knowledge of this topic is only based on Folland's Real Analysis and Modern Techniques, Chapter2. I would appreciated some help. Prove $$\cal{E} = \big\{ \{a_1\} \times \{a_2 \} \times ... \times \{a_n\} \times \{0,1\} \times \{0,1\} \times ... : n \geq 1, a_1,..., a_n \in \{0,1\} \big\} \bigcup \{\emptyset\}$$ is an elementary family on {0,1}^N and the $\sigma$-algebra generated by it is the Borel $\sigma$-algebra on $\{0,1\}^N$ is based on the metric $$d((x_n)_{n \geq 1}, (y_n)_{n \geq 1}) = \sum_{n \geq 1} \frac{\lvert x_n - y_n \rvert}{2^{n}}$$