Obviously, if $x$ has a finite decimal representation, it will remain finite after cancelling the finite number of digits, hence it will still be rational. If $x$ has an infinite decimal representation, then it's either purely periodic od impurely periodic (what's the english term?). a) If it's purely periodic, then assume the period length $T$. The we have $a_{k+(nT+j)r}=a_{k+jr}$, hence the digits are cancelled in a periodic manner, with the period length $rT$, leaving a periodic decimal number. b) It it's impurely periodic, then the periodic part starts after first $l_0$ digits, hence it's enough to analyze the digits $a_{k+nr}$ such that $k+nr>l_0$, and the conclusion will be similar to the one in a).