""Convince" seems to have replaced "persuade" almost completely in UK ... "I convinced him to come with us to the cinema""
"It depends how much persuading he needed :-)"
Exactly. I'd reluctantly accept "I convinced him to give himself up to the police", but would prefer "I convinced him that he should give himself up". Going to the cinema may indeed require hard persuasion, but I still feel "convince" would be wrong in this kind of "argument".
IMHO, you convince people that a situation is as you say it is, or a course of action is the right one but you don't (well, didn't!) follow it with an infinitive. Otoh, you do follow "persuade" with an infinitive - not invariably, but most of the time.
BS