How does one select some $i$ to prove $\exists i$ s.t. $s_i \leq A$ where $A$ is the average of real numbers $(s_1 + s_2 + . . . + s_n) / n$
It seems like a trivial proof, but writing a proof for such an existence conventionally does not seem obvious to me.
Perhaps I am reading it wrong, but I am sure it says something like: For any real number $A$ which is the average of a set, $S$, of real numbers, there exists a number $s \in S$ s.t. $s$ is less than or equal to $A$.
Is there a special way to choose $i$ to prove this existence? It seems I cannot trivially select a set of numbers and show there is a value that satisfies $s_i \leq A$ since it is some arbitrary set. How would you select $i$ for this proof?