it is one of the homework questions, it doesnt show the proof, just the solution

im not sure where you are jumping from if .999... = 1 then 1=2

.999... is an infinite decimal
.999...9 would mean there is an actual end to it, thus a terminating decimal, thus it would not be equal to 1

if you cant see that 1/3 = .333... repeating then there is some other assumption you are making that isnt valid, repeating decimals are just as valid a number as an integer or a fraction, they arent approximations or rounding

edit - page 1.999... owned