What is meant by "infinitely often" in this problem on Borel sets?

Question

As the title suggests, I am having some trouble understanding an exercise regarding Borel sets. In particular I am trying to

Show that the set of real numbers that have a decimal expansion with the digit $5$ appearing infinitely often is a Borel set.

My question here is what is meant by the phrase "infinitely often" in this case? I have never seen this phrase before in a formal context. This attempt of mine is horrific so don't butcher me, but I tried starting out in the following way:

Let $\mathcal{A}_5$ denote the set of real numbers that have a decimal expansion with the digit $5$ appearing infinitely often, i.e. every $a\in\mathcal{A}_5$ can be written as $$a=\ldots+ 5\cdot10^n+5\cdot10^{n-1}+\ldots+5\cdot 10 +5+5\cdot10^{-1}+\ldots 5\cdot10^{-m-1}+5\cdot10^{-m}+\ldots$$

Am I interpreting "infinitely often" correct here? If not, what is it trying to say?

Answer 1

The phrase means just this; if $$a_k 10^k + a_{k-1} 10^{k-1} + \dots + a_0 + a_{-1} 10^{-1} + a_{-2} 10^{-2} + \cdots$$ is the decimal expansion of a real number, then the set $$ \{ n : a_n = 5, -\infty < n \leq k \} $$ is an infinite set.

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Pulled up from the comments below: to elaborate, the given condition means that there is some infinite subsequence of $5$'s in the sequence $( a_n )$.

Take care that it does not mean that all $a_n$'s are $5$, or that all $a_n$'s after a certain point are $5$. For example, you could have that $a_n = 5$ whenever $n$ is even and $a_n = 7$ whenever $n$ is odd. This is a number whose decimal expansion contains infinitely many $5$'s, but it also contains infinitely many non-$5$'s.

Answer 2

"Infinitely often" means "infinitely many times". In other words, of all of the digits in the decimal expansion of your number, infinitely many of them are $5$. It doesn't mean that all of them are $5$, or even that "most" of them are $5$ in any sense, just that there are infinitely many $5$s.

Answer 3

The requisite concept for the problem of infinite 5s in the decimal expansion of a real number is the limit supremum of a sequence of sets: $$ S=\limsup_{n \to \infty} A_n = \bigcap_{n \geq 1} \bigcup_{j \geq n} A_j. $$ The meaning of $S$ is that $x \in S$ iff x is in infinitely many $A_j$. A useful suggestion by Robert Israel as to how to construct the $A_j$ is here. My quibble with the answer is that the $A_j$ can be constructed using open intervals. The endpoints can be excluded since they have a decimal expansion with a finite number of 5s.