First, m = n；; Otherwise, if X tends to infinity, f(x) tends to infinity or 0 tends to infinity, because the periodic function must have points A (a) = k and pT+a(p is a positive integer) when P tends to infinity, but f(pT+a)=f(a) does not tend to infinity. As in the case of approaching 0, there is always a proposition that f(a) is not equal to 0, otherwise it is equal to 0. It is proved that the coefficients corresponding to the numerator and denominator are proportional: the numerator is multiplied by a number to make their first coefficient equal (this will not affect the periodicity of the function), and this fraction can be changed into 1 plus a true fraction, which requires that f(x) is a periodic function and tends to 0 when x tends to infinity, which is contradictory (also proved in the above way).