Let $(\Omega,\mathcal{F})$ be a measurable space. The following are equivalent:
$\ X:\Omega \to \mathbb{R} $ is a Borel function.
$\{\omega\in\Omega:X(\omega)>a\}\in\mathcal{F}$ for all $a\in\mathbb{R}$.
$\{\omega\in\Omega:X(\omega)< a\}\in\mathcal{F}$ for all $a\in\mathbb{R}$.
$\{\omega\in\Omega:X(\omega) \in B\}\in\mathcal{F}$ for all open subsets $B\subset\mathbb{R}$.
$\{\omega\in\Omega:X(\omega) \in B\}\in\mathcal{F}$ for all closed subsets $B\subset\mathbb{R}$.
How on earth would I prove this? I have no idea where to start. Any help would be very much appreciated. Thanks