The price of an American option is given by $V_n = \max\{G_n, \frac{1}{1 + r}(pV_{n +1}(H) + qV_{n + 1}(T)\}$, where $p$, $q$ are the risk neutral probabilities.
I have two questions.
How can one intuitively see that this must be the formula to avoid arbitrage? If possible cite a trivial example showing arbitrage if one does not take the maximum of these two values.
How to intuitively see that the ideal time to exercise the option is $\min\{n: V_n = G_n\}$ ?
Thanks.