Prove the result in equation (9.4). As in the text we use $c$ and $p$ to denote the European call and put option price, and $C$ and $P$ to denote the American call and put option prices. Because $P \geq p$, it follows from put-call parity that
$$
P \geq c+K e^{-r T}-S_0
$$
and since $c=C$,
$$
P \geq C+K e^{-r T}-S_0
$$
or
$$
C-P \leq S_0-K e^{-r T}
$$
For a further relationship between $C$ and $P$, consider
Portfolio I: One European call option plus an amount of cash equal to $K$.
Portfolio J: One American put option plus one share.
Both options have the same exercise price and expiration date. Assume that the cash in portfolio $I$ is invested at the risk-free interest rate. If the put option is not exercised early portfolio $\mathrm{J}$ is worth
$$
\max \left(S_T, K\right)
$$
at time $T$. Portfolio $\mathrm{I}$ is worth
$$
\max \left(S_T-K, 0\right)+K e^{r T}=\max \left(S_T, K\right)-K+K e^{r T}
$$
at this time. Portfolio I is therefore worth more than portfolio J. Suppose next that the put option in portfolio $\mathrm{J}$ is exercised early, say, at time $\tau$. This means that portfolio $\mathrm{J}$ is worth $K$ at time $\tau$. However, even if the call option were worthless, portfolio I would be worth $K e^{\tau \tau}$ at time $\tau$. It follows that portfolio $\mathrm{I}$ is worth at least as much as portfolio $\mathrm{J}$ in all circumstances. Hence
$$
c+K \geq P+S_0
$$
Since $c=C$,
$$
C+K \geq P+S_0
$$
or
$$
C-P \geq S_0-K
$$
Combining this with the other inequality derived above for $C-P$, we obtain
$$
S_0-K \leq C-P \leq S_0-K e^{-r T}
$$