0.999 ... is a repeating decimal , which means the digit `` 9 '' is repeated forever . This decimal yields a quick proof for 0.999 ... = 1 . Multiplication of 9 times 1 produces 9 in each digit , so 9 Ã -- 0.111 ... equals 0.999 ... and 9 Ã -- equals 1 , so 0.999 ... = 1 :