Show that, if $C$ is the price of an American call with strike price $K$ and maturity $T$ on a stock providing a dividend yield of $q,$ and $P$ is the price of an American put on the same stock with the same strike price and exercise date, then
$$S_{0} e^{-q T}-K \leqslant C-P \leqslant S_{0}-K e^{-n}$$
where $\delta_{0}$ is the stock price, $r$ is the risk-free interest rate, and $r>0$. (Hint: To obtain the first half of the inequality, consider possible values of:
To obtain the second half of the inequality, consider possible values of: