Distinct-positive-integer subset-sum lower bound
= Distinct-positive-integer subset-sum lower bound
{title2=$|\{\sum_{i\in A}s_i:A\subseteq[n]\}|\geq1+n(n+1)/2$}
For $0<s_1<\cdots<s_n$, adding the largest integer creates at least $n$ new <sums of subsets>. If $P=\sum_{i<n}s_i$ is the previous maximum, the values $P+s_n$ and $P+s_n-s_i$ for $i<n$ are distinct new sums above $P$. Induction gives the displayed bound. Equality holds for $s_i=i$, whose subset sums fill the interval of integers from zero to $n(n+1)/2$.