mid 16th century (in the sense 'overcome, defeat in argument'): from Latin convincere, from con- 'with' + vincere 'conquer'. Compare with convict
Convince used (with an infinitive) as a synonym for persuade first became common in the 1950s in the US, as in she convinced my father to branch out on his own. Some traditionalists deplore the blurring of distinction between convince and persuade, maintaining that convince should be reserved for situations in which someone’s belief is changed but no action is taken as a result (he convinced me that he was right) while persuade should be used for situations in which action results (he persuaded me rather than he convinced me to seek more advice). In practice the newer use is well established.