Understanding a real analysis question Let $x\in(0,1]$ show that there is a unique sequence $\alpha_1,\alpha_2,\ldots$ with the following properties:


*

*$\alpha_i=0$ or $1$

*$\alpha_1=1$ for infinitely many $i\in \textbf{N}$

*$\displaystyle\sum_{i=1}^{\infty}\frac{\alpha_i}{2^{i}}=x$


What is this asking? Please refrain from answers, I just want to know what the problem is asking and where I should begin?

It's asking to show that every number in the unit interval has a unique binary representation, but there is a switcheroo: all the numbers that would ordinarily terminate by ending in an infinite run of zeros must end in an infinite run of ones instead.  $0$ is excluded for the same reason that $1$ is excluded in the ordinary version of the problem.
