Is there an easy way to see that binary expansion is unique? Let $n \in \mathbb{N}$. Using the Euclidean algorithm, it is straightforward to see that every natural number can be written as
$$n = \sum_{j=0}^m \epsilon_j(n) 2^j $$
where $\epsilon_j(n) \in \{0,1\}$. 
Is there an easy way to show that this way of writing the number is unique?

How do you know the ordinary base $10$ expansion is unique?
Suppose digit strings $s$ and $t$ both represent the positive integer $n$. Then the units digit of each must be $d = n \pmod {10}$. So you can lop off both units digits. The lopped strings then both represent $(n-d)/10$.
Continue with the other digits (from the right) until you're done. Or, for a formal induction proof, apply that argument to the least $n$ with two representations to deduce a contradiction.
This argument works for any base. It's the standard algorithm for base conversion, finding the digits from right to left. It depends on knowing that you can do division with remainder, but not on the full strength of the Euclidean algorithm.
