A Proof of Theorem 1 is guaranteed in Π

For the first argument, we use induction. For the second part, we it is essentially a Coupon Collector's problem. Consider a general sampling problem: for any finite set N with |N | = N. For any n, whose sampling probability is p(c), with a probability at least 1 δ, it requires at most log(1/δ) for n to be sampled. We consider the combination lock problem [20].