Let $n,k$ be any positive integers. Given real numbers $0\leq a_1,\dots,a_n\leq C$, is it always possible to partition them into $k$ blocks, where each block contains consecutive elements of the sequence, so that the sums of any two blocks differ by at most $C$? If not, what is the best bound?
Remark that this difference may need to be as large as $C$, such as when $a_1=C$ and $a_2=\dots=a_n=0$. An idea is to start with any partition and try to reduce this difference, but since each block needs to contain consecutive elements it is not clear how far the difference can be reduced.
Edit: A difference of $2C$ can be achieved, as shown by Alex Ravsky below. So the bound is between $C$ and $2C$.