I'm reading a solution to a problem that I don't fully understand. The statement writes out the binary expansion of a real number and requires that there is no natural number $N$ beyond which the binary expansion is $1$ forever.
I'm not able to understand (a) why such a binary expansion exists or (b) why this implies that the resulting binary expansion is unique. I'm not sure if there's an analogy with the ability to choose a decimal expansion for any real number that doesn't terminate in $9$'s.
I'd appreciate if someone could help me understand this.