She argued 'convincingly' [='persuasively'] that more money should be given to public schools.
There is no 'convincing' evidence to support his theory.
Your argument/story isn't very 'convincing'.
(..) We stop fighting our enemies, the world will die. What of it? Then it'll be out of its misery. You know how you sound? Like a man who's convincing himself of something he doesn't believe. Each of us has a destiny. For good or for evil. I get the point. I wonder if you do. (..)1942 Casablanca