I attached a picture for more clarity. Think of each box as a column of a vertical histogram and its cookies as its height (i.e. $y$-coordinate). Let's draw for each column a line $s_n$ parallel to the $x$-axis from the top of the column until he reaches the $y$-axis, intersecting it at $t_n$. Serge's process is to collect the cookies drawing a line for each $y$ and then counting the elements that this line cuts on the columns. We notice that each plate made between $t_{n-1}$ and $t$ has the same number of cookies. So each $(t_{n-1},t_n)$ corresponds to the class of equal elements found between the two lines. We also notice that if two columns have the same height (i.e. the same number of cookies) then we can delete one of them without affecting the interval design on the $y$-axis (to each of those columns corresponds a same $s_k$). Then, WLOG, each column has a different height. It's clear that if there are $n$ columns, we will have $n$ distinct values on the $y$-axis. Hence, the result. [geogebra]da39a3ee5e6b4b0d3255bfef95601890afd80709[/geogebra]