Superincreasing sequence ($w_k>{\sum }_{j<k}{w}_j$)

A positive sequence is superincreasing if each term exceeds the sum of all earlier terms. If $w_{k+1}>2w_k$, a geometric series proves ${\sum }_{j<k}{w}_j<w_k$. Distinct finite subsets of its indices have distinct sums: cancel their common terms and compare the largest remaining term with the sum of all smaller terms. This gives uniqueness of minimum spanning trees and of simple graph paths with fixed endpoints when a finite weighted graph has such separated edge weights.