Show that the set of real numbers that have a decimal expansion with the digit 5 appearing infinitely often is a Borel set.
Let $E$ be the set of all real numbers that have a decimal expansion with the digit 5 appearing infinitely often. To show that $E$ a Borel set, first we need to verify that $E$ is a $\sigma$-algebra. First to show that $E$ is closed under countable intersections, take $r_1, r_2, \dots \in E$ and note that their intersection will be the infinite subsequence of 5s. Thus the countable intersection is in $E$. Similarly for countable unions, the countable union contains the infinite subsequence of 5s so it is in $E$ too. Thus $E$ forms a $\sigma$-algebra.
I am wondering if my approach is good in first showing that $E$ is $\sigma$-algebra and then showing that it is a Borel set. Assuming that I showed correctly that $E$ is a $\sigma$-algebra, I am not sure on how to show the last step, that $E$ is a Borel set. I think the direction I have to go in is to translate between the decimal expansion of real numbers and how they can be written as open subsets.
Hint: The set $E_1$ of real numbers that have a decimal expansion where the first digit after the decimal point is $5$ is the union of $[n+ 0.5, n+0.6]$ and $[-n-0.6, -n-0.5]$ for $n = 0, 1, 2, \ldots$ ($0.6$ is included because $0.6 = 0.5999\ldots$). This is a closed set: its complement is open. Generalize to $E_k$ involving the $k$'th digit. Express your set as a countable intersection of countable unions of these.