Don't prove people wrong. It doesn't help anyone, 99% of the time. The other 1% is important, but you can usually tell a life or death situation apart from a normal one.
Two things happen when you prove someone wrong. It hurts his ego, which is the worst thing to jab. Suddenly, and this is automatic, so don't take it too personally, his impression of you takes a hit. The second is, it boosts your ego. No one will ever thank you for proving them wrong, which is kind of what you're expecting, so you're left only with disappointment, and this air of importance that doesn't have to be there. Plus, when you realize what you said only made things awkward and the other person mad or taken aback, you'll only wish you kept quiet.
Nothing happens when you prove someone wrong. No one comes out and shakes your hand and hands you a medal. All you do is make someone feel and/or look stupid, which although some (many) people deserve, isn't right most of the time, and more importantly makes you look like a horrible person, so even purely for selfish reasons, it's best you don't.
These are different from arguments. Civilized arguments are awesome. They dig deep at the matter and you might actually come out of it with something better, even if you get a bit riled up (not too much, and don't show it too much if you do). What I'm talking about is when you know, 100%, that what someone said or claimed is wrong and it doesn't have anything to do with you personally really, they're just wrong, and you get this burning in your chest to say something because the world shouldn't be filled with false information like this. (This is different from intentional lies, don't let those slide). But does it really matter who said that? Where this and that is located, and who made this or that? What that's made of, or how that works? 99% of the time, no. So smile and nod and think of pleasant things, because you'll forget it in five minutes and you'll be a better person because of it.
Now, you can still use these opportunities to your advantage other than to inflate your own sense of self worth, which doesn't feel that great anyway. Use it as a gauge. I've found that people who get things wrong, get things wrong a lot. They love to talk, to boast, to claim, and you just facepalm yourself (internally) whenever you hear something that makes you realize this person doesn't know what the hell he's talking about, and the worst thing is he doesn't know that he doesn't know what he's talking about.
Then there are the people who don't say as much, but instead ask more questions, are more curious, they know what they don't know, and they are consistently right, because they don't claim to know things they know they don't know much about.
Seek these people out. They're awesome. You learn that these are the people to ask questions to and to seek advice from. Their words, even their thoughts, weigh heavier. And try to stay away from the other kind. These are the people you never ask about anything to, and you take what they say with a grain of salt. Besides, it relieves you the stress of constantly having to bang your head against the desk.