Find a number in a set that is not equal to sum of any other numbers

Question

Given a set of nonzero real number S= $\{d_1,\cdots, d_N\}$, can I find a number $d_e \in S$ such that there does not exist $d_m,d_j\in S$ such that $d_e = d_m+d_j$ ? Assume $m$ and $j$ can be equal and $m,e,j$ denote the indices with $e\neq m,j$.

Answer 1

This is true for any set of real numbers with a positive minimum. If $S\subseteq\Bbb R$ and $0<x\in S$ such that for all $y\in S, y\ge x$, then for any $y,z\in S$, $x\le y<y+z.$

This is false for the set $S=A\cup(A+1)$ where $\{-2,1\}\subseteq A$. If $x\in S$, then $x\in A$ or $x-1\in A$. In the first case, $x+1\in S$, so $x=(x+1)+(-1)$ is a decomposition, otherwise $x-1\in S$, so $x=(x-1)+1$ where $1\in S$.

Answer 2

Not always.

Let $S= \{1, 2, 2^2, ..., 2^N\}$ and consider the case when $d_e = 1$.