I remember I was stunned by my teacher demonstration (I was 14)
It was SO simple :

X=.999999999...
10X=9.9999999999...
10X-X=9
9X=9
X=1

Needless to say that since this day there were no doubts for me...
.999999... is 1

UPDATE : I should have read the tilly's link first.
My demonstration is presented there, apologizes for the redundant information...

"Only Bad Coders Code Badly In Perl" (OBC2BIP)