bound on the size of set whose $k$-term subset sums are different

Question

I'm trying to solve the following problem from additive combinatorics:

For $k \geq 2$ integer. Let $A \subset \{1,\dots,N\}$ be such that all $k$-term sums from $A$ are different, then $$|A| \leq (k \cdot k! \cdot N)^{1/k}$$