A Gazillion Is Equal To Zero

Algebra Level 3

Here's my proof that 10000000000infinitely many 0’s1\underbrace{000000000\ldots0}_{\text{infinitely many 0's}} is equal to 0.
In which of these steps did I first make a flaw in my logic?

Step 1: Let X=9999999999infinitely many 9’sX = \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} .

Step 2: Divide both sides by 10, X10=9999999999infinitely many 9’s.9  .\dfrac{X}{10} = \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} .9 \; .

Step 3: Take the difference between these two equations:

XX10=9999999999infinitely many 9’s9999999999infinitely many 9’s.99X10=9999999999infinitely many 9’s(9999999999infinitely many 9’s+0.9)0.9X=9999999999infinitely many 9’s(9999999999infinitely many 9’s+0.9)0.9X=0.9X=1 \begin{aligned} X - \dfrac{X}{10} &=& \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} - \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} .9 \\ \dfrac{9X}{10} &=&\underbrace{999999999\ldots9}_{\text{infinitely many 9's}} - ( \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} + 0.9) \\ 0.9X &=& \xcancel{\underbrace{999999999\ldots9}_{\text{infinitely many 9's}}} - ( \xcancel{\underbrace{999999999\ldots9}_{\text{infinitely many 9's}}} + 0.9) \\ 0.9X &=& -0.9 \\ X &=& -1 \end{aligned}

Step 4: Substitute back the value of XX and add 1 to both sides to the equation.

9999999999infinitely many 9’s=19999999999infinitely many 9’s+1=1+110000000000infinitely many 0’s=0\begin{aligned} \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} &=& -1 \\ \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} + 1&=& -1+1 \\ 1\underbrace{000000000\ldots0}_{\text{infinitely many 0's}}&=& 0 \\ \end{aligned}

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