convince
To make someone believe or agree by giving good reasons.
To convince someone means to persuade them that something is true or that they should do something. When you convince your parents to let you get a pet, you give them good reasons why it's a sensible idea: you'll take care of it, you're responsible enough, and having a dog would teach you important lessons. When a lawyer convinces a jury, she presents evidence and arguments that make them believe her client's side of the story.
Convincing requires solid reasons, clear explanations, or compelling evidence. You need to show why something makes sense or is worth doing. A scientist might convince other scientists of a new discovery by showing careful experiments and data. A friend might convince you to try a new game by explaining why it's fun and letting you watch them play.
Notice that you can't force someone to be convinced. Real conviction happens when people genuinely change what they believe because of the strength of your reasoning. Someone who is convinced has been truly won over, not bullied or tricked.