Consider the family of sequences of the form $.012\ldots n$ for any natural number $n$. So, the sequences in this family are: $.01, .012, .0123, .01234,$ etc.
Now consider to manipulate each sequence in this way:
1) Start from the rightmost digit; let's say that the rightmost digit is $n$;
2) If $n$ is even, replace $n$ by $0$ and add $n/2$ to the digit immediately on the left;
3) if $n$ is odd, replace $n$ by $1$ and add $(n-1)/2$ to the digit immediately on the left;
4) keep on modifying the sequence on the left until the only digits are '$0$' and '$1$'.
E.g., suppose that you consider the sequence $.01234.$ Then you modify it in this way: $.01250; .01410; .03010; .11010$.
I want to show that, for any $n$, after rewriting the sequence with only '$0$' and '$1$', I will never obtain any sequence that starts in this way $x.yy$'$y$'... where $x\neq 0$. So, e.g., I will never obtain any sequence like $1.0101$.
This problem is probably analogous to show that any sequence of the form $(0*2^{-1})+(1*2^{-2})+(2*2^{-3})+...+(n*2^{-(n+1)})$ is $<1$. However, I am not sure how to prove it.