"; "The teenager's parents were not swayed by their daughter's arguments" [v] win approval or support for; "Carry all before one" convince v] make (someone) agree, understand, or realize the truth or validity of something; "He had finally convinced several customers of the advantages of his product" Seeing the definitions, I'd say "convince" tends to resort more to the brain, to the logical abilities. But they're pretty much similar to me...
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