If 0.999…. wasn’t 1, then 1 – 0.999… isn’t 0. Then you can divide by it. Then, 1 / (1 – 0.999…) is a number that is bigger than all of the natural numbers (natural numbers are the counting numbers: 1, 2, 3, …). However, every real number must be smaller than some natural number (round the number up). This is a contradiction.
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