Question

prove that a binary American put will always be valued more than a binary European put

Answer 1

It can be proved by Time value of money concept. In American put case, you get the option to sell a stock at a given strike price. If you exercise this option at t=0, you receive the strike price at t=0 and can invest it at the risk-free rate.

Lets imagine the risk free rate is 4% and the strike price is 10$. this means at t=1, you would get 10$ plus interest amount. If, on the other hand, you did'nt exercise the option early, at t=1 you would simply receive the strike price (10$). Basically, the strike price, which is your payoff for a put option, doesn't earn interest.