There are two possible scenarios that could arise from the situation that you have framed in the question. First is where such a person is someone who finds it difficult to understand things. Such a person would be adamant about his opinions because he is not having complete knowledge or understanding of the issues in focus.
I do not say that the person is egotist rather I would say that he is somewhere unable to look at things from different points of view in different situation. In such a case, with slight efforts, you will be able to explain your points satisfactorily to that person and the best thing is that with some efforts, that person would succeed in understanding the situation completely and eventually take a correct stand of his own. It is important to note that it is easy to place facts in front of such a person only because that person is able to put his ego aside and accept his mistake.
Second is a scenario wherein the opposite person is actually an egotist. Such a person has a false belief of self-importance. He is more self-centred and purposely does not want to understand the scenario from every perspective. His main idea is to enforce his opinion on others with an idea of satisfying self-ego.
In this scenario, first of all you need to be careful that you do not lose your temper. I am saying this because there are high chances that the opposite person’s adamant attitude will force you to lose your cool. Hence, before you speak up anything calm yourself down. Then one by one place the facts in front of that person, explaining him each fact in detail and how it contradicts his ideas or opinions.
With a smart and cool approach, you might succeed in making that person realize the truth. But if after all the efforts that person is not ready to leave his stand, then you should leave the topic positively with the wish that he understands the truth some day. We can explain things to a lot of people. But it is always upon them to implement it in their attitude and character. Every person is independent. We can only wish for their betterment. Rest is upon them.
Also, a point of caution here is that if the opposite person is not ready to listen to you then learn to let go off that issue right there. If you become adamant that the other person must listen to you, then you would be fuelling your ego. So learn to let go off things at the right point. This is for your own good.