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If $A\subset\mathbb{N}$ is a finite set of integers all of whose subset sums are distinct then \[\sum_{n\in A}\frac{1}{n}<2.\]
This was proved by Ryavec. The stronger statement that, for all $s\geq 0$, \[\sum_{n\in A}\frac{1}{n^s} <\frac{1}{1-2^{-s}},\] was proved by Hanson, Steele, and Stenger [HSS77].

See also [1].