Borel sigma algebra of extended Real Line

Question

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.)

Answer 1

Hint:

For the first inclusion ($ \subset $): $\mathcal{B}(\overline{\mathbb{R}})=\sigma( \left\{ ]a,+\infty] \ / \ a\in \mathbb{R} \right\}) \ $ and $\ ]a,+\infty] \cap \mathbb{R}=]a,+\infty[ \in \mathcal{B}(\mathbb{R})$

For the second inclusion ( $\supset $): $E=E\cap \overline{\mathbb{R}}=\left ( E\cap\mathbb{R} \right )\cup \left ( E\cap \left\{ -\infty , +\infty \right\} \right )$ and $ E\cap \left\{ -\infty , +\infty \right\} \in \mathcal{P} \left ( \left\{ -\infty , +\infty \right\} \right ) \subset \mathcal{B}(\mathbb{\overline{R}}). $

Can you finish ?