Problem: $(x_i)_{i=1}^n$ is a finite sequence of positive integers.
Define $f\big(S\big)=\displaystyle \sum_{i\,\in\, S\,\subseteq\, [n]}x_i$, and suppose $f$ is injective. Prove that: $$\sum_{i=1}^n \frac{1}{x_i}<2$$
Any ideas on how to tackle this problem?
Try showing that $x_i\geq 2^{i-1}$ for all values of $i$. Then the inequality comes from the fact that $$\sum_{i=1}^\infty \frac1{2^{i-1}} = 2$$