Here's my proof that 1infinitely many 0’s000000000…0 is equal to 0.
In which of these steps did I first make a flaw in my logic?
Step 1: Let X=infinitely many 9’s999999999…9.
Step 2: Divide both sides by 10, 10X=infinitely many 9’s999999999…9.9.
Step 3: Take the difference between these two equations:
X−10X109X0.9X0.9XX=====infinitely many 9’s999999999…9−infinitely many 9’s999999999…9.9infinitely many 9’s999999999…9−(infinitely many 9’s999999999…9+0.9)infinitely many 9’s999999999…9−(infinitely many 9’s999999999…9+0.9)−0.9−1
Step 4: Substitute back the value of X and add 1 to both sides to the equation.
infinitely many 9’s999999999…9infinitely many 9’s999999999…9+11infinitely many 0’s000000000…0===−1−1+10
Your answer seems reasonable.
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