Prove $\sum_{m=i}^{n}2^{n-m}\binom{m}{i}=\binom{n+1}{i+1}+\ldots+\binom{n+1}{n+1}=\sum_{m=i}^{n}\binom{n+1}{m+1}$ without induction
Hint: Consider organizing the subsets by what the $(i+1)$th element is.
Hint: Consider organizing the subsets by what the $(i+1)$th element is.