Contempt undermines any rational argument you might make to convince another person of anything. People are not dumb, they can sense when the underlying tone is disparaging. The message gets lost in some interpretation that sounds like “you’re a bad person because you don’t believe my idea” which leads to defensiveness.
The way to convince people is to care. Genuinely caring about what the other person thinks and says can’t be faked. Nor can one’s objectivity and sense of fallibility. When you care, you naturally become more open to receiving other ideas (not necessarily believing, just not outright rejecting them) and so does the other person.
Next time you are trying to convince someone of something important, ask yourself, is this coming from a place of contempt?