My grandparents strongly dislike my girlfriend. I’m clueless.

I just got into a long distance relationship with my girlfriend who’s in California right now. We’ve been chatting, video calling and talking to each other for months and we’ve finally decided to make things official. I decided to tell my family about it but my grandparents strongly disagree. They’re saying things like “women from USA are cheaters and inconsistent. You’ll just get your heartbroken in the long run.” I can see that they have good intentions since they’re just worried about me but I really am confident about this girl. We’ve only met once during my stay as a foreign exchange student program. I’ve gotten to know her personally and I can say she’s amazing. How can I convince them that there’s nothing to worry about?