(assuming $n$ is finite)
This seems like an easy proof, but how could one write it down nicely?
I was thinking about proving it by cases: if one $s_i$ is negative, say $s_k$ than the statement is trivially true, with $S>s_k$,
If all $s_i$ are positive, then the statement is also true because the sum is greater than the parts, but I don't know how one would write this formally? (i.e. it seems like it follows from the properties of adding positive numbers together, but I don't know how one could explain this using symbols?)
Alternatively, is there a different way to prove this beside by two cases?