In Paul Wilmotts quantitative finance books he says that the the value of an American option satisfies the following
$$ \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2 \frac{\partial^2V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV \leq 0 \quad \quad (1) $$ and by the no arbitrage condition we have
$$V(S,t) \geq P(S,t) \quad \quad (2)$$ where P is the time dependent payoff. Now to quote the justification he gives for the price of American and European call options to be same is that
"If we substitute the Black–Scholes European call solution, in the absence of dividends, into the inequality (1) then it is clearly satisfied; it actually satisfies the equality. If we substitute the expression into the constraint (2) with P (S, t) = max(S − E, 0) then this too is satisfied. The conclusion is that the value of an American call option is the same as the value of a European call option when the underlying pays no dividends "
My question is why cant the same be said about American put options .As far as I understand the European put also satisfies (1) with equality .