> So we are back to "that's the way is 'cause I said so". I can live with > that, but the implication is that the demonstration follows from > something more fundamental.
That makes it sound very arbitrary. It's more a case of "that's the way it is because mathematicians have determined that is what is most sensible and thus they say so". That determining was the culmination of the hard thought of numerous people and you can learn the details from history books such as Hairer & Wanner's 'Analysis by its history'.
-- The world will little note, nor long remember what we say here Lincoln at Gettysburg