It can be transformed into a proposition that the fraction is either divisible (that is, finite decimal) or infinitely cyclic decimal.

For a fraction a/b, if it is divisible, it is a finite decimal.

If you can't divide it all the time, it will definitely produce a remainder every time. According to the pigeon hole principle, there must be the same remainder after dividing by b+ 1 times. This will start the cycle from the same two remainders. In other words. The number of digits in the cyclic part of cyclic decimal must be less than the divisor B.

For example, 1/7 =0. 142857 cycle.

That is to say, all fractions are finite decimals or infinite cyclic decimals. It cannot be an infinite acyclic decimal.