protective put strategy involves long equity and long put option. My question is as follows:

strike is K, and current stock price is S. Amercian put option is always worth more than the corresponding European put option (under same strike and maturity), since American put option is sometimes worth its intrinsic value K-S, it must hold true that the corresponding European put is sometimes worth less than its intrinsic value K-S, so does it mean that when the European put is worth less than the intrinsic K-S, I implement the protective put by buying the European put with premium p (less than its intrinsic K-S), then the worst case would be I only make profit of (K-S)-p?

Many thanks!