Sum of $n$ positive real numbers $a_1, ...,a_n$ is $1$. Let $S_k$ be maximal sum of k distinct elements of $a_n$. (they can be equal but must have different indexes). What is $\sup S_k$ and $\inf S_k$ among all sequences of length $n$? For $k=1$ or $k=n-1$we can use Pigeonhole Principle. How to handle other cases? What if numbers are nonnegative?
Please provide a hint unless solution is non-trivial. Thanks!
HINT:
Say you get the choose the largest $k$ numbers out of $n$ numbers.( think amounts of money). The total amount is $\$1000$. What is the worst case scenario for you? If it makes is easier, take $n=10$, and $k=3$.