Let $\{f_n\}$ be a sequence of Baire one functions from $\mathbb{R}$ to $\mathbb{R}$ in which $f_n(\mathbb{R})$ is isolated in $\mathbb{R}$ for every $n$.
Hence the $H_n = \{x: |f_n(x) - f_{n-1}(x)| <3. 2^{-n}\}$ is a Borel set of the additive class one. My tutor said that $\mathbb{R} - H_n$ is also a Borel set of the additive class one. Why? It should be true using fact that $f_n(\mathbb{R})$ and $f_{n-1}(\mathbb{R})$ are isolated. But I do not how to prove it.