Just like what the title says.
Does: $$n = n.\overline 01?$$ For example, $1.\overline 01 = 1$?
Similar to $(n-1).\overline 9 = n$, for example, $0.\overline 9 = 1$.
The last statement is true but intuitively I also feel the first is true as well but I look for a proof online and couldn't find one.
I am oversimplifying the notation here obviously just to make it easier to understand on what I mean. The idea came form if $3.\overline 9 = 4$ then does coming form the other direction of 4 i.e. $5$ still equals $4$ which in this case would be $4.\overline 01$?
Some people have argued in the comments that the notation 1.000...01 cannot represent a number, because it makes no sense to have a remaining 1 at the end of an unending sequence of repeating zeroes.
I will not answer the question "does this notation represent a number?", but I will attempt to prove that if this notation corresponds to a number, and if the notation plays nice with arithmetic manipulations, then this number must be equal to the number 1.
Let $$x = 1.000...01$$ (whatever that means).
Then: \begin{align*} x & = \,\,\, 1.000...01 \\ 10 x & = 10.000...10 \\ 10 x & = 10.000...010 \\ 10 x & = \,\,\,9 + x \\ 9 x & = \,\,\,9 \\ x & = \,\,\,1. \end{align*}
A word of caution: If you can give a more formal definition of the meaning of the notation 1.000...01, then you'd need to verify that my calculations above are still correct with your definition.
In particular, between the second and third lines I appeared to have used something along the lines of "an unending sequence of zeroes followed by a 10 is the same thing as an unending sequence of zeroes followed by a 1". Is this really true?