Representation of integers

Strong induction. Base case $n=1$: $1=2^0$. ✓

Inductive step. Suppose every integer $<n$ has a binary expansion.
- If $n$ is even: $n=2m$ with $m<n$, so $m=\sum 2^{a_i}$; then $n=\sum 2^{a_i+1}$, distinct powers.
- If $n$ is odd: $n-1$ is even and $<n$, so $n-1=\sum 2^{b_i}$; then $n=2^0+\sum 2^{b_i}$, with $0$ distinct from all $b_i$ (since $n-1$ even means all $b_i\ge1$).

Conclusion. Every positive integer is a sum of distinct powers of $2$.