I'm trying to prove the following (where $\overline{\mathbb{R}} = \mathbb{R} \cup \{+\infty, -\infty \}$):
The Borel sigma algebra $\mathcal{B}(\overline{\mathbb{R}})$ is the sigma algebra generated by open sets in $\overline{\mathbb{R}}$, where $(\overline{\mathbb{R}}, \arctan)$ is a metric space. Prove that $\mathcal{B}(\overline{\mathbb{R}}) = \{ E \subset \overline{\mathbb{R}}: E \cap \mathbb{R} \in \mathcal{B}(\mathbb{R})\}$.
I think I've managed to do $\sigma(\{ \text{open sets in } \overline{\mathbb{R}}\})\subset \{ E \subset \overline{\mathbb{R}}: E \cap \mathbb{R} \in \mathcal{B}(\mathbb{R})\}$, but I'm not so sure it is correct. Moreover, I'm very stuck on $\{ E \subset \overline{\mathbb{R}}: E \cap \mathbb{R} \in \mathcal{B}(\mathbb{R})\} \subset \sigma(\{ \text{open sets in } \overline{\mathbb{R}}\})$. Does anyone have a good solution for this?
(There is a similar answer in Topology and Borel sets of extended real line, but I'm looking for an answer without using a Topology explicitly. This answer doesn't achieve that.)