Here's nice little proof (well, it's not laid out formally and it uses inequalities that really should be proved first (though they're easy inequalties to prove), but I've gauged it exactly for the sort of person who will argue 0.99.... is not equal to 1. Though are alot better and more comphrehensive proofs, people who've I've shown this one seem to liek it) that I came up with:
(Firstly just about every crank whose claimed that 0.999.. is not equal to 1 claims that there are nbo numebrs inbetween 0.999.. and 1, this proof really sets out to deal with this claim)
if x = 0.999... then there are 3 possible relations that x and sqrt(x) could have
1. sqrt(x) > 1
Howver if this is the case then 0< x < 1 must be UNTRUE as for any 0 < x < 1, 0 < sqrt(x) < 1. If this is the case x > 1
2. sqrt(x) = 1
If this is the case then x = 1 is the only solution, meaning that 0.999 does indeed equal 1.
3. sqrt(x) < 1
If this is the case then there is a number between 1 and x (as if 0 < x < 1 then sqrt(x) > x); what is the decimal representation of this number? Of cousre at this point someone could argue that in this case sqrt(x) = x, but this means that x msut equal 1 or zero).