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1. It is false. 0.999... just isn't 1, literally.
2. And (0.9 x / x<0.1)≠1 Λ (0.9 x / x<0.1)<1.
3. No amount of repetition of 0.0XXX…. could ever equal 0.1, which is what is necessary to add to 0.9 in order to equal 1.
4. A house built on sand cannot divide itself.
5. For the complete proof, visit here.
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