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Bob is doing his calculations in base 2 (aka. binary): $$0.111..._2 = 1_2$$ similarly to the the following in base 10: $$0.999..._{10} = 1_{10}$$ The apparently wrong part is correct when the calculation is done in base 2: $$10_2 - 1_2 = 1_2$$ The last sentence states that every number is in base 10, which interpreted correctly means that every number is in base $$10_2 = 2_{10}$$