Where is the mistake in this proof about rationals having eventually periodic decimal expansions?

Here is a thing you can always do but which people don't seem to teach: if you have a proof that you're suspicious of, you can go through the proof with an example. At some point, you'll write down a false statement about your example, and that's probably where the mistake is.

(Of course you can't do this if the proof is a proof by contradiction because, if the thing you're trying to prove is true, there shouldn't be any examples. This is one reason to avoid proofs by contradiction.)

Let's take a really simple example, namely $x = \frac{1}{2} = 0.500 \dots$, whose decimal expansion is clearly eventually periodic but not periodic. $[x] = \frac{1}{2}$ again. But $[10^r x] = 0$ for all $r \ge 1$. So this is where the mistake is. (As Hagen von Eitzen says, what pigeonhole will actually tell you is that there exist $0 \le r \neq s \le q$ such that $[10^r x] = [10^s x]$. It does not guarantee that either $r$ or $s$ is equal to $0$, but there's a second mistake here: you also aren't guaranteed that this fractional part will be nonzero!)

"Then by the pigeonhole principle, $\{10^rx\}=\frac aq$" is a non-sequitur. The principle says that among too many pigeons, two must be in the same hole, but it does not state thet two pigoens must be in the first hole.