Show $\{y \mid \limsup\limits_{i \rightarrow \infty}d(x_i,y) \leq 1\}$ and $\{y \mid \liminf\limits_{i \rightarrow \infty}d(x_i,y) \leq 1\}$ are Borel

Question

I was struggling immensely to show that $\{y \mid \limsup\limits_{i \rightarrow \infty}d(x_i,y) \leq 1\}$ and $\{y \mid \liminf\limits_{i \rightarrow \infty}d(x_i,y) \leq 1\}$ are Borel. My first attempt consisted of falsely writing:$\{y \mid \limsup\limits_{i \rightarrow \infty}d(x_i,y) \leq 1\}=\bigcap\limits_{N=1}^{\infty}\bigcup\limits_{n \geq N}\{y \mid d(y,x_n) \leq 1\}$ and $\{y \mid \liminf\limits_{i \rightarrow \infty}d(x_i,y) \leq 1\}=\bigcup\limits_{N=1}^{\infty}\bigcap\limits_{n \geq N}\{y \mid d(y,x_n) \leq 1\}$. However, people were giving me counterxamples to show they are not true. My question is, how can I properly write the sets $\{y \mid \limsup\limits_{i \rightarrow \infty}d(x_i,y) \leq 1\}$ and $\{y \mid \liminf\limits_{i \rightarrow \infty}d(x_i,y) \leq 1\}$ as union/intersections of open/closed sets in order to show they are Borel?

Answer 1

Note that $$\limsup\limits_n d(x_n, y)\leq 1$$ $$\Leftrightarrow \forall \epsilon>0,\text{ } \exists N\in \mathbb{N} \text{ such that } d(x_n,y)\leq 1+ \epsilon \text{ for }n\geq N$$ $$\Leftrightarrow \forall k\in \mathbb{N},\text{ } \exists N\in \mathbb{N} \text{ such that } d(x_n,y)\leq 1+ 1/k \text{ for }n\geq N.$$

The set of $y$'s satisfying the last requirement can be expressed as $\bigcap\limits_{k=1}^\infty \bigcup\limits_{N=1}^\infty \bigcap\limits_{n=N}^\infty \{y: d(x_n, y)\leq 1+ 1/k\}$.

EDIT: The most efficient way to show that the sets are measurable is to note that, for any $n$, the function $y \mapsto d(x_n, y)$ is continuous, hence measurable, and then that the $\limsup$ of measurable functions is measurable. Then the set $\{y: \limsup\limits_nd(x_n,y)\leq 1\}$ is just the pre-image of $(-\infty, 1]$ under a measurable function.