To make someone believe, or feel sure about something, especially by using logic, argument or evidence
make (someone) agree, understand, or realize the truth or validity of something; "He had finally convinced several customers of the advantages of his product"
If someone or something convinces you to do something, they persuade you to do it. That weekend in Plattsburgh, he convinced her to go ahead and marry Bud. = persuade
If someone or something convinces you of something, they make you believe that it is true or that it exists. Although I soon convinced him of my innocence, I think he still has serious doubts about my sanity The waste disposal industry is finding it difficult to convince the public that its operations are safe