= Solution
Let $f_n$ be Borel measurable and converge pointwise to the finite real value $f$. For every real $a$,
$$
\boxed{\{f>a\}=\bigcup_{\substack{r\in\mathbb Q\\r>a}}
\bigcup_{N\ge1}\bigcap_{n\ge N}\{f_n>r\}}.
$$
If $f(x)>a$, choose a rational $r$ strictly between them; convergence puts $f_n(x)>r$ eventually. Conversely eventual strict inequality $f_n(x)>r>a$ implies $f(x)\ge r>a$. The right side belongs to $\mathcal E$ by its countable union/intersection operations. Since these half-lines generate the <Borel sigma-algebra>, $f$ is a <Borel measurable function>.
Back to article page
