>>9390356

>>9390334

>>9390332

>>9390319

>>9390313

[eqn]0.\dot01=a=0.c_1c_2c_3...c_n (c_k \in \{0,1,2,3,4,5,6,7,8,9\})[/eqn]

[eqn]

0=0.\dot0=b=0.d_1d_2d_3...d_n (d_k \in \{0,1,2,3,4,5,6,7,8,9\})

[/eqn][eqn]

\text{if }a \ne b \text{ is true}

[/eqn][eqn]

\text{then }c_final \ne d_final\text{ is true}

[/eqn][eqn]

0=c_1=c_2=c_3=...=c_\infty=d_final\text{ but }c_final = 1,\text{ thus } c_\infty \ne c_final

[/eqn][eqn]

\text{Let's say }c_final = c_{\infty +1}

[/eqn][eqn]

\mathbb{C}=\in\{c_1,c_2,c_3,...,c_{\infty + 1}\}, \mathbb{D}=\in\{d_1,d_2,d_3,...,d_\infty\}

[/eqn][eqn]

n(\mathbb{C})=\infty +1, n(\mathbb{D})=\infty,

[/eqn][eqn]

\text{and } n(\mathbb{C}) \ne n(\mathbb{D})

[/eqn][eqn]

\text{so } \infty \ne \infty +1 [/eqn]

But it isn't(https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel),

[eqn]

\text{thus }\infty=\infty+1[/eqn][eqn]

n(\mathbb{C})=n(\mathbb{D}), c_final=c_{\infty+1}=c_\infty

[/eqn][eqn]

\text{ but }c_\infty=0\text{ so }c_final \ne d_final\text{ is not true}[/eqn]

[eqn]\text{thus }a \ne b \text{ is also not true}[/eqn]

[eqn]\therefore 0.\dot01 = 0[/eqn]