Then again, perhaps you have a different understanding of the word "proving"...
Exactly; the action of "proving" obviously does not involve getting someone to see something
which they are intent to not see. If it did, nothing would ever be proven, nothing could ever be resolved, and victory would always go to the stubborn rather than the most realistic.
Mature minds have an independent viewer function which can be persuaded by a well crafted argument, even when that argument goes against what they want to be true. Lesser minds, when confronted by facts which challenge their biases simply refuse to acknowledge the facts.
Realistically, if you aren't persuadable by another person's argument, they you aren't doing anything other than posturing. Inflexibility doesn't prove that you're right; it proves that you aren't really functioning at a serious level.