Read: how to actually change someone's mind.

The below three readings are all links to research papers related to how to engage and persuade those who disagree with you.

You can't "prove someone wrong."  When presented with news articles or reports that directly contradict their beliefs, many people will exhibit a "backfire effect" -- doubling down on their beliefs and entrenching them even more strongly.  This is currently on display in an extreme form as Trump calls any critical reporting "fake news."

Previous research in political science has found that it is possible to change issue opinions by directly providing relevant facts to subjects (Kuklinski et al. 2000; Gilens 2001; Howell and West 2009). However, such authoritative statements of fact (such as those provided by a survey interviewer to a subject) are not reflective of how citizens typically receive information. Instead, people typically receive corrective information within ‘‘objective’’ news reports pitting two sides of an argument against each other, which is significantly more ambiguous than receiving a correct answer from an omniscient source. In such cases, citizens are likely to resist or reject arguments and evidence contradicting their opinions—a view that is consistent with a wide array of research (e.g. Lord et al. 1979; Edwards and Smith 1996; Redlawsk 2002; Taber and Lodge 2006).
In this paper, we report the results of two rounds of experiments investigating the extent to which corrective information embedded in realistic news reports succeeds in reducing prominent misperceptions about contemporary politics. In each of the four experiments, which were conducted in fall 2005 and spring 2006, ideological subgroups failed to update their beliefs when presented with corrective information that runs counter to their predispositions. Indeed, in several cases, we find that corrections actually strengthened misperceptions among the most strongly committed subjects.
Brendan Nyhan and Jason Reifler (2010)

Liberals and conservatives have different fundamental values, making it hard to craft persuasive arguments.  It's better to appeal to those values, not your own, if you want to change someone's mind.  And even more persuasive than that are the arguments of a conservative -- someone who holds their same values.

MFT researchers surveyed thousands of people around the world regarding the concerns that were morally relevant to them (Graham et al., 2009; Graham et al., 2011). Results led to the proposal of five primary moral foundations—harm/care, fairness/reciprocity, ingroup/loyalty, authority/respect, and purity/sanctity. Importantly, this research finds that liberals tend to endorse foundations based on caring and protection from harm (harm) and maintenance of fairness and reciprocity (fairness) more strongly than conservatives.  However, conservatives tend to endorse moral concerns related to ingroup-loyalty (loyalty), respect for authority (authority), and protection of purity and sanctity (purity) more than liberals. Related research largely supports these findings (Caprara et al., 2006; Jost, Glaser, Kruglanski, & Sulloway, 2003; Lakoff, 2002; Thorisdottir et al., 2007), and together, this body of work indicates that liberals possess stronger moral convictions related to fundamental aspects of harm and fairness (e.g., benevolence, nurturance, equality, social justice), and conservatives possess stronger convictions related to group loyalty, authority, and purity (e.g., patriotism, traditionalism, strictness, religious sanctity).
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The results of our studies suggest that arguments appealing to the messenger’s values rather than the audience’s values were not only unpersuasive to the target audience but also did not impact the attitudes of those already in favor of the political position being argued for. For instance, conservatives presented with a military spending argument grounded in loyalty and authority concerns did not show any greater support for military spending than conservatives presented with an argument grounded in fairness concerns.
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We found that the morally reframed arguments conservatives wrote in favor of official English were more persuasive to the liberal targets than the other messages, and this difference was mediated by the extent to which liberal participants reported that the message resonated with their moral values more.
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Although our results demonstrate the efficacy of framing moral arguments in terms that appeal to the values of those on the other side of the political spectrum, the method could have a variety of unintended consequences. Appealing to the other side’s values might reinforce or validate those values (e.g., Lakoff, 2002). For instance, if liberals reframe support for same-sex marriage in terms of the more conservative moral principles, their arguments may legitimize those principles, facilitating their use to support conservative positions in the future. Relatedly, advocates might be reluctant to use moral reframing because they feel doing so compromises their own beliefs, a position apparently taken by a small number of participants in the above auxiliary study. Furthermore, moral reframing could lead advocates to assimilate to, or soften in their views of values they previously opposed.
Matthew Feinberg and Robb Willer (2015) and supported by further research in 2016.

In an extended argument (and in this case, specifically confined to the Internet), the first few people to argue for a position are more likely to be successful, perhaps because extended debate only hardens beliefs.  They also found that conversations with people who were specifically willing to engage on an issue didn't display the "backfire effect" discussed above, but being too intense was still a problem.

We find (example above to the contrary) that a challenger that enters the fray before another tends to have a higher likelihood of changing the OP’s opinion; this holds even for first-time CMV challengers, and so is not a trivial consequence of more experienced disputants contriving to strike first. Although engaging the OP in some back-and-forth is correlated with higher chances of success, we do not see much OP conversion in extended conversations. As for opinion conversion rates, we find that the more participants there are in the effort to persuade the OP, the larger the likelihood of the OP changing her view; but, interestingly, the relationship is sublinear.
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Style features and interplay features both prove useful and outperform a strong baseline that uses bag-of-words. In particular, interplay features alone have strong predictive power, achieving an improvement of almost 5% in accuracy over the baseline method (65.1% vs 59.6%) in a completely fresh heldout dataset. Our results also show that it is useful to include links as evidence—an interesting contrast to studies of the backfire effect: “When your deepest convictions are challenged by contradictory evidence, your beliefs get stronger." However, it hurts to be too intense in the counterargument. The feature with the most predictive power of successful persuasion is the dissimilarity with the original post in word usage, while existing theories mostly study matching in terms of attitude functions or subject self-discrepancy.
Chenhao Tan, Vlad Niculae, Cristian Danescu-Niculescu-Mizil, and Lillian Lee (2016) (PDF)