We have distributed two hundred balls into one hundred boxes with the restrictions that no box got more than one hundred balls, and each box got at least one. Prove that it is possible to find some boxes that together contain exactly one hundred balls.
Assume to the contrary that no class of boxes contains $100$ balls. If there is a box with exactly $2$ balls, there is no class of boxes with $98$ balls, which implies that there is no class of boxes with $96$ balls. Continuing this way we see that there is no class containing $4$ balls, so there is no class containing $2$ other than the initial box with $2$ balls, i.e, this box is unique.
Now, if there is a box with $1$ ball, then there is no class of $99$ balls and continuing this way, there is no class of $2$ balls so no box with $1$ ball other than our initial box with $1$ ball. So again, this box is unique. Thus, in the best scenario we have our balls distributed as $1,2,3,...,3$ but this is more than $200$ balls, contradiction.
What do you think? Is this a valid proof now?