I have the following problem.
Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ be Borel-measurable functions. Show that $$\{x|\limsup\, f_n(x)=\infty\}, \,\,\{x|\limsup \,f_n(x)=c\}, c\in \mathbb{R}$$ are borel sets.
From the lecture we just know that since $f_n$ are Borel-measurable, then also $g(x)=\limsup \,f_n(x)$ is Borel-measurable. I only looked at the second set for the moment and I thought if $g(x)$ is measurable, we get that $g^{-1}(c)=\{x|g(x)=c\}\subset B(\mathbb{R})$. Remark that $c\in B(\mathbb{R})$. But I'm not sure if this works because somehow I found it too easy.