Please solve all parts for thumbs up!

Consider an American put and a European put with the same strike price K and expiration T. In this problem, we investigate the relationship between the price P of the American put and the price p of the European put. We already know that Pp0 since an American put is always worth at least as much as a European put. Consider a portfolio where you sell one American put and buy one European put and invest the proceeds Pp. Note that the initial value of the portfolio is zero. (a) Show that if r=0, then the portfolio is guaranteed to be worth at least Pp at time T. Deduce that P=p. Hint: Draw a timeline from 0 to T. Plot the different cash flows under different scenarios. (b) Show that if r>0, then the portfolio at time T will be worth at least (Pp)erT+KKer(Tt) where t is either the time when the American put is exercised or it equals T. Since er(Tt)erT, conclude that Pp+K(1erT), or else there is an arbitrage opportunity