This is a proof that the decimal representation of any rational number terminates with an infinitely repreating sequence of digit(s). Let p/q be any rational number, where p and q are integers in decimal representation. and p and q have no common factor. The significant case is where p is less than q.

Consider the process of constructing the decimal representation of the quotient of p divided by q by the long division algorithm. The first step of that algorithm is of the form

                          d
                        _____________________
                     q | p.0000000000000000000...
                         a b
                         ___
                           r

The remainder is some integer between 0 and (q-1). If the remainder in any step is 0 then the quotient terminates in an infinite sequence of zeroes. Suppose zero never occurs as a remainder. If any remainder occurs a second time the sequence of digits from its first occurrence to its second occurrence will be repeated in the quotient ever afterwards. Since there are only (q-1) possible remainders a repetitious must eventually occur.

For proof that any decimal representation terminating in
a repeating decimal is a rational number see
Repeating Decimal Representations