Prove that, if [math]A_1, A_2, \ldots,A_n[/math] are independent events defined on a sample space [math]\Omega[/math] and if [math]0 \lt P(A_j) \lt 1[/math] for all [math]j[/math], then [math]\Omega[/math] must have at least [math]2^n[/math] points.