Let X_n be a binomial negative random variable with n degrees of freedom or Binomial (p, -n) with 0(or equal) 2 we have:

n/(n+X_n) >(or equal) (n-1)/(n-1+X_n) and E[(n-1)/(n-1+X_n)] = 1-p

Now show that if Y_1,Y_2,Y_{3, ... , is a sequence of i.i.d. Geometric(p) random variables, then we can simulate} X_{n as:}

X_{n =} Y₁ ... Y_n

Finally, show this implies that:

lim (n-->inf) (n/n+X_n) = lim (n-->inf) (n-1)/(n-1+X_n) = 1-p