Here's my proof that \(1\underbrace{000000000\ldots0}_{\text{infinitely many 0's}} \) is equal to 0.

In which of these steps did I **first** make a flaw in my logic?

**Step 1**: Let \(X = \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} \).

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

**Step 3**: Take the difference between these two equations:

\[ \begin{eqnarray} X - \dfrac{X}{10} &=& \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} - \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} .9 \\ \dfrac{9X}{10} &=&\underbrace{999999999\ldots9}_{\text{infinitely many 9's}} - ( \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} + 0.9) \\ 0.9X &=& \require{cancel} \xcancel{\underbrace{999999999\ldots9}_{\text{infinitely many 9's}}} - ( \xcancel{\underbrace{999999999\ldots9}_{\text{infinitely many 9's}}} + 0.9) \\ 0.9X &=& -0.9 \\ X &=& -1 \end{eqnarray} \]

**Step 4**: Substitute back the value of \(X\) and add 1 to both sides to the equation.

\[\begin{eqnarray} \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} &=& -1 \\ \underbrace{999999999\ldots9}_{\text{infinitely many 9's}} + 1&=& -1+1 \\ 1\underbrace{000000000\ldots0}_{\text{infinitely many 0's}}&=& 0 \\ \end{eqnarray} \]

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