The following problem is from Royden & Fitzpatrick (4 ed.). I am stuck on showing (ii), can someone please help me prove it? Thank you.
$\def\R{{\mathbb R}}$ Page 59, problem 8. (Borel measurability) A function $f$ is said to be $\textbf{Borel measurable}$ provided its domain $E$ is a Borel set and for each $c,$ the set $\{x\in E | f(x) > c\}$ is a Borel set. Verify that Proposition 1 and Theorem 6 remain valid if we replace "(Lebesgue) measurable set" by "Borel set." Show that: (i) every Borel measurable function is Lebesgue measurable; (ii) if $f$ is Borel measurable and $B$ is a Borel set, then $f^{-1}(B)$ is a Borel set; (iii) if $f$ and $g$ are Borel measurable, so is $f\circ g;$ and (iv) if $f$ is Borel measurable and $g$ is Lebesgue measurable, then $f\circ g$ is Lebesgue measurable.
$\textit{Proof.}$ Every Borel measurable set is Lebesgue measurable since $B\in B(\R),$ then $B$ is a Lebesgue measurable set except perhaps on a set of measure $0.$ For (iii), assume $g: \mathbb{R} \to \mathbb{R}$ and $f: \mathbb{R} \to \mathbb{R}.$ Then, $(f\circ g)^{-1}((c,\infty)) = g^{-1}\circ f^{-1} ((c,\infty)).$ By the hypothesis, $f^{-1}((c,\infty)) = B\in B(\R).$ By definition of Borel set, any member of $B(\R)$ is the result of countable set operations or a member of the topology on $\R.$ Any member of the topology on $\R$ may be written as the countable result of set operations on $(a,\infty)$ for some $a\in \R,$ so $g^{-1}(B) \in B(\R).$ Thus, $f\circ g$ is Borel measurable. Now to prove (iv), assume $f: (X,T) \to (\R,U)$ with $(X,T)$ a general topological space, and $U$ the standard topology on $\R.$ By definition, any Borel set $B\in B(\R)$ is a result of countable set operations as an open set. Now given that $f^{-1}((c,\infty)) \in B(x),$ any open set may be written in terms of open rays and any Borel set in $\R$ can be written in terms of these open sets. Hence, the inverse image of a Borel set in $\R$ is the countable set theoretic result of operations on $f^{-1}((c,\infty))$ which is a Borel set as $B(x)$ is a $\sigma$-algebra.