Uniqueness of non terminating binary decimals in $[0,1]$ How do we prove that non terminating binary decimals has a unique binary expansion? I notice that this question has been posted before but no answer has been given yet...Thanks for your explanation!
 A: If I understand correctly you want to prove that every real number has a unique non-terminating expansion.
Suppose $a$ has two non-terminating binary expansions, since they are different they must be different in at least one of the digits, consider the leftmost digit that is different.
This exists since the numbers left of the decimal point are a finite amount, so we can label the leftmost digit 1, then 2, then 3 and so on, Therefore the digits in which they differ can be seen as a subset of the natural numbers, and every subset of the natural numbers has a minimum element 
Let $b$ be the number which has the binary representation of the first digits up to the one that is different,if we consider the "big" representation of $a$ we have $a>b$ since it must have other non-zero digits after (otherwise it would terminate) on the other hand if we look at the "small" representation of $a$ we get $b\geq a$ since $a$ has the same representation as $b$ up to a point but then it has a smaller digit. therefore $a>b\geq a\implies a>a$, a contradiction.
