Are sets of point convergence of Borel functions Borel?

Question

Given a metric space $X$, $x_0\in X$ and a sequence of Borel functions $f_n\colon X\to X$, let us define

$$A = \{x\in X\colon \lim_{n\to\infty} f_n(x) = x_0\}. $$

Is there a way to see that $A$ is a Borel subset of $A$? (Especially, if we know that poitwise limit of Borel functions are Borel.)

Answer 1

Let $d$ be a metric on $X$, let $g_n(x) = d(f_n(x),x_0)$. Since $d(\cdot,\cdot)$ is continuous on $X\times X$, $g_n$ is Borel. Let $g=\varlimsup g_n$. Your $A$ is $g^{-1} (\{0\})$.

Answer 2

\begin{align*} A&=\bigcap_{n}\bigcup_{N}\bigcap_{k\geq N}\{x\in X:f_{k}(x)\in B_{1/n}(x_{0})\}, \end{align*} and $f_{k}^{-1}(B_{1/n}(x_{0}))$ is Borel.