Sampling without replacement helps, slightly

With $N = 1{,}000$ applications and $K = 100$ lots, compute $P_5$ from (0.6) exactly and compare it with the independent-draw approximation $1 - (1-p)^5$. Then prove that $P_m \ge 1 - (1-p)^m$ for all $m$ (the inequality in Proposition 0.5), and explain in one sentence why it holds.