Induction series : Something is not right

Prove that any positive integer $n$ can be written as distinct powers of $2$ .

1. Base case: $1 = 2^{0}, 2 = 2^{1}, 3=2^{0}+2^{1},.....,7=2^{2}+2^{1}+2^{0}$
2. We assume that statement holds true for positive integers $1,2,3,.........n$
3. For $n+1$ , let $n+1=2^{k}+a$ , where $0<a<n+1$
4. Let $a=2^{x_{1}}+2^{x_{2}}+2^{x_{3}}+.......+2^{x_{m}}$ , with $x_{1},x_{2},.....x_{m}$ all being distinct whole numbers.
5. Because $n+1=2^{k}+2^{x_{1}}+2^{x_{2}}+2^{x_{3}}+.....+2^{x_{m}}$ , we can conclude that $n+1$ can also be the sum of distinct powers of 2.
6. Hence all positive integers can be represented as distinct powers of 2.

Between which steps am I missing something? If the missing step is between 1 and 2, input as 12.