Bob claims to have a proof that $0.\dot1=1$.
That's $0.\overline1=1$, $0.(1)=1$ or $0.11111...=1$ in other common formats.
The proof starts $$\text{If }1x=0.\dot1,\\
\text{then }10x=1.\dot1\\
10x-1x=1.\dot1-0.\dot1\\1x=1\\
\text{replacing the value of }1x\text{ for }0.\dot1\text{ (as defined at the start)}\\
\\0.\dot1=1$$$$\text{If }1x=0.\dot1,\\
\text{then }10x=1.\dot1\\
10x-1x=1.\dot1-0.\dot1\\1x=1\\
\text{substituting in the value of }1x\text{ for }0.\dot1\text{ (as defined at the start)}\\
\\0.\dot1=1$$
He is not wrong (Ignore the title). Everything is correct. Every number in this question is in base $10$.
How is this possible?
