Infinite sequence has $m$ consecutive digits – regular languages

Question

Suppose we have an infinite sequence over the decimal alphabet, call it $w$. For $k\in \{0,1,2,3,4,5,6,7,8,9\}$, let $L_k = \{m\geq 0: \text{$w$ contains a run of $m$ $k$'s}\}$ be a language. Here a run denotes a contiguous block of the sequences. Prove that $L_k$ is a regular language.

Here is my idea. Clearly, if $w$ contains arbitrarily long runs of $k$, then $L_k$ is pretty much just the set of all positive natural numbers, which is regular. But if $w$ has a maximum run of $k$'s, then $L_k$ is a finite set, which means it's regular. This is as far as I got, this language "feels" like it shouldn't be regular, but I am told it is. Would the regularity of $L_k$ even depend on whether it is possible to compute whether $w$ has a maximum run of $k$'s?

Answer 1

You don't explain how the input to $L_k$ is encoded, but $L_k$ will be regular under any reasonable encoding.

Indeed, if $w$ contains a run of $m$ many $k$'s, then it also contains a run of $m'$ many $k$'s for any $m' \leq m$. Therefore either $L_k$ is finite (indeed, of the form $\{0,\ldots,m\}$) or it consists of all of $\mathbb{N}$.

You seem to be troubled by the fact that $L_k$ is not necessarily computable from (a description of) $w$. But there is absolutely no such requirement – all we have to show is that for each $w$, the languages $L_k$ are regular. We are not claiming that we can compute $L_k$ from (a description of) $w$.