Ok please, for once and for all, ENLIGHTEN US!
To be quite honest I can't remember the exact explanation! !!
We were shown this at uni but have since forgotten what the lecturer said. I've come across this 'proof' a few times since and its driven me nuts trying to figure it out each time!
All I remember is that it is to do with differentiating an infinite series, 'x lots of x', which is dodgy unless it converges.
Apologies that this is not really an adequate explanation but at least I am now not alone in being driven nuts!
P.S. Some of you here are at uni aren't you? Maybe you could ask one of your lecturers?