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'.
(..) Horace. Merlin's beard! You don't need to disfigure me, Albus. Well I must say you made a very convincing armchair, Horace. It's all in the upholstery. I come by this stuffing naturally. What gave me away? Dragon's blood. (..)2009 Harry Potter and the Half-Blood Prince