Consider the function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined such that $\forall x\leq y :f(x)\leq f(y)$. My lecture notes state that $f$ is Borel measurable with the seemingly simple proof:
$$\forall a\in\mathbb{R}:f^{-1}((a,\infty))=[b,\infty)\quad \text{or}\quad f^{-1}((a,\infty))=(b,\infty),$$ where $b=\inf f^{-1}((a,\infty))$. Since $[b,\infty)$ and $(b,\infty)$ are Borel sets, $f$ is a Borel measurable function.
Generally, to show that $f:X\rightarrow Y$ is Borel measurable, we have to show that $f^{-1}(B)$ is a Borel set for any Borel set $B\subset Y$. However, the proof above is restricted to Borel sets of the type $(a,\infty)$. So how can it be a valid proof?
In general, when you deal with functions $f$ from $(X,\mathscr{A})$ to $ (\mathbb{R},\mathbb{B})$, it's equivalent to show only that $f^{-1}(]a,\infty[)\in \mathscr{A}$ for all $a\in \mathbb{R}$. It comes from the fact that $\mathbb{B}$ is generated by $\{]a,\infty[,a\in\mathbb{R}\}$.