I have found two distinct proofs for the problem:
Show that the set of real numbers that have a decimal expansion with the digit 5 appearing infinitely often is a Borel set.
Both of which I could not prove some steps.
Proof 1) Let $E:= \{ x \in \mathbb{R}: \text{$x$ has a decimal expansion with the digit 5 appearing infinitely often} \}$, and let $E_n := \{ x \in \mathbb{R} : \text{$x$ has a decimal expansion with the digit 5 appearing in the $n$th position}\}$. We show that $\displaystyle E = \limsup_{n \to \infty} E_n = \bigcap_{n \geq 1} \bigcup_{m \geq n} E_m$, so that $E$ is a Borel set as long as $E_n$ is a Borel set.
I could not prove why $\displaystyle E = \limsup_{n \to \infty} E_n = \bigcap_{n \geq 1} \bigcup_{m \geq n} E_m$, and cannot proceed with the proof.
Proof 2) Let $E_n := \{ x \in \mathbb{R} : \text{$x$ has a deciaml expansion with the digit $5$ appearing only $n$ times}\}$. Then $E = \mathbb{R} \backslash (\bigcup_{n=0}^{\infty} E_n)$, hence $E$ is a Borel set as long as $E_n$ is a Borel set. (This $E$ is the same as in proof 1)
I could not prove why $E_n$ is a Borel set, or why $E_n$ is closed.
Any corrections regarding the proofs will be appreciated!