Yes, Wiesel does convince his audience.
He explains what indifference is and how it is different from hatred. Indifference, he says, is worse. He explains how indifference played a part in some of the greatest atrocities in the twentieth century. He contrasts these atrocities with some of the good works achieved during the century, works which were achieved through action, not through inaction.
Therefore, Wiesel proves that inaction and indifference lead to tragedy, while action and compassion lead to greater peace. In order to prevent the horrors of the twentieth century from repeating themselves in the future, one must turn aside from indifference.