I don't understand why the distance between root(n, N) and root(n, N+1) is the probability that N is a perfect nth power. What if we choose to compute the dista

I don't understand why the distance between root(n, N) and root(n, N+1) is the probability that N is a perfect nth power. What if we choose to compute the distance between root(n, N) and root(n, N+0.5) ? By the same logic, if N is a perfect nth power, there exists one integer in the interval [root(n, N),root(n, N+0.5)], and since the distance between all consecutive integers is one, the probability is the ratio etc ... And what about 0.2? 0.1? We can keep going until the probability becomes zero ! And this can't be !